title: Asymptotic stability of solitary waves for the $1d$ NLS with an attractive delta potential
authors: Satoshi Masaki Jason Murphy Jun-ichi Segata
text: 
  
  Stability of solitary waves]Asymptotic stability of solitary waves for the 1d NLS with an attractive delta potential
  
  S. Masaki]Satoshi Masaki
  Department of Systems Innovation, Graduate school of Engineering Science, Osaka University
  masaki@sigmath.es.osaka-u.ac.jp
  
  J. Murphy]Jason Murphy
  Department of Mathematics & Statistics, Missouri University of Science & Technology
  jason.murphy@mst.edu
  
  J. Segata]Jun-ichi Segata
  Department of Mathematical Sciences, Kyushu University 
  segata@math.kyushu-u.ac.jp
  
  
  We consider the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases.  We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result <cit.> beyond the regime of small solitary waves and extend the results of <cit.> from orbital to asymptotic stability for a suitable family of solitary waves. 
  
  
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  § INTRODUCTION
  
  We consider the one-dimensional nonlinear Schrödinger equation (NLS) with an attractive delta potential.  This equation has the form
  
      i∂_t u = Hu + f(u), 
      
      u|_t=0=u_0.
  
  
  
  
  Here H=-12∂_x^2 + qδ_0, where q<0 and δ_0 is the Dirac delta distribution (see Section <ref>).  The nonlinearity in (<ref>) is assumed to be of the form f(u)=σ |u|^p u, where σ∈{± 1} and p>4 (the mass-supercritical regime).  The choice σ=+1 yields the defocusing equation, while σ=-1 gives the focusing equation.  This equation is related to a simple model describing the resonant nonlinear propagation of light through optical wave guides with localized defects <cit.>.
  
  Equation (<ref>) admits an explicit one-parameter family of solitary wave solutions of the form u(t,x)=e^iω tQ_ω(x), where
  
      HQ_ω+ω Q_ω + f(Q_ω) = 0
  
  (see Section <ref>).  Here the frequency ω is restricted to (0,12q^2) in the defocusing case and to (12q^2,∞) in the focusing case, with 12q^2 corresponding to the eigenvalue for the underlying linear operator.  The stability properties of these solitary waves have been studied in previous works <cit.>.  In the mass-supercritical case p>4 with defocusing nonlinearity, all solitary waves are orbitally stable.  In the focusing case, there exists Ω∈(12q^2,∞) such that one has stability for ω∈(12q^2,Ω) and instability for ω∈[Ω,∞).  The frequency Ω arises as the unique frequency at which ddωQ_ω_L^2^2=0.  
  
  In our previous work <cit.>, we showed that one has asymptotic stability for all solitary waves with frequency ω sufficiently close to 12q^2.   In this regime, the solitary waves arise as small multiples of the eigenfunction for the underlying linear equation, and the fact that the solitary waves are small plays a key role in the proof.
  
  In this work, our main goal is to establish asymptotic stability beyond the regime of small solitary waves for the model (<ref>).  We consider initial data of the form
  
      u_0 = Q_ω_0 + ṽ_0,
  
  where Q_ω_0 is an orbitally stable solitary wave and ṽ_0 is small in H^1 and in a weighted L^2 space.  Our main result (Theorem <ref> below) is an asymptotic stability result for suitable choices of ω_0.  In particular, we will show that if a certain spectral condition holds at the frequency ω_0 (see Definition <ref> below), then the solution to (<ref>) with initial data u_0 as in (<ref>) scatters to a nearby solitary wave as t→∞.  Before stating the theorem precisely, let us briefly introduce the general approach and explain the origin of the spectral condition that plays a key role in the argument.
  
  Orbital stability (established in <cit.>) guarantees that for initial data of the form (<ref>), the corresponding solution admits a decomposition of the form 
  
      u(t) = e^iβ(t){Q_ω_0 + ṽ(t)}
  
  for some phase β(t), where ṽ(t) is small in H^1 for all t≥ 0.  One might first try to establish asymptotic stability by proving that in fact, ṽ(t) decays/scatters as t→∞.  To study the behavior of ṽ(t), one can first use (<ref>) and (<ref>) to derive a nonlinear evolution equation for ṽ.  The hope would then be to use this equation together with the smallness assumption on ṽ_0 to run a bootstrap argument that yields decay estimates for ṽ.  Such an argument would require estimates for the linear part of the evolution equation, which involves the matrix operator
  
      Ł(ω_0)=[[      0  H+ω_0; -H-ω_0      0 ]]+σ Q_ω_0^p[[      0      1; -(p+1)      0 ]]
   
  arising from the linearization around the solitary wave. 
  
  At this point, the spectral properties of the operator Ł(ω_0) come into play.  Using (<ref>), we can immediately find two independent vectors in the generalized kernel of Ł(ω_0) (namely, [0 Q_ω_0]^t and [∂_ω Q_ω|_ω=ω_0 0]^t), which implies the existence of a two-dimensional subspace on which the linear evolution e^tŁ(ω_0) does not decay.  Moreover, it is possible for Ł(ω_0) to have additional purely imaginary eigenvalues, even when Q_ω_0 is orbitally stable.  Thus we see that to obtain decay estimates, it would at least be necessary to restrict to the continuous spectral subspace of Ł(ω_0). On the other hand, there is no reason a priori that the perturbation ṽ(t) above should belong to this subspace; indeed, the decomposition (<ref>) incorporates only a single modulation parameter.  Thus it is not clear whether or not the component ṽ(t) actually decays as t→∞.
  
  In light of the above observations, we modify the approach as follows.  We firstly restrict attention to frequencies ω_0 so that the dimension of the generalized kernel of Ł(ω_0) is exactly two and such that Ł(ω_0) has no other eigenvalues.  This is part of the spectral condition appearing below. Next, instead of (<ref>), we look for a decomposition of the form
  
      u(t) = e^iΦ(t){Q_ω(t)+v(t)},
  
  where now both the phase Φ(t) and the frequency ω(t) may vary in time.  In particular, we will choose these two modulation parameters to guarantee that v(t) belongs to the continuous spectral subspace of Ł(ω(t)) for each t≥ 0.  This requires that we impose suitable orthogonality conditions (see (<ref>) below), which we can accomplish by making use of the existing orbital stability results and the implicit function theorem (see Proposition <ref>).  We then essentially carry out the strategy outlined above, namely, we use dispersive estimates for the evolution group and a bootstrap argument to derive decay estimates for the perturbation v(t).  We must also derive suitable estimates for the modulation parameters themselves.
  
  Before stating the main result, we must discuss one additional technical issue that is also connected to the spectral condition stated below.  In the bootstrap estimates for both v and the modulation parameters, we must exhibit enough decay to integrate in time.  The lowest order term in the modulation parameters will be quadratic in v, while the equation for v will still ultimately contain some linear terms (due to a change of variables that serves to remove the time-dependence from the underlying linearized operator).  This means that we must do better than the typical |t|^-1/2 decay rate associated to dispersive estimates for one-dimensional Schrödinger-type equations.  By utilizing weighted norms, we can obtain an estimate with an integrable decay rate.  However, this requires that the linear operator Ł obeys a further spectral condition, namely, the absence of a resonance at the edge of the continuous spectrum. 
  
  We may now state the main spectral assumption and our main result.  We recall the definition of Ω given above (see also Section <ref>).
  
   Let ω∈(0,12q^2) in the defocusing case and ω∈(12q^2,Ω) in the focusing case. We say that ω satisfies the spectral condition if Ł(ω) has a generalized kernel of dimension 2, no other gap or embedded eigenvalues, and no resonances.
  
  
   Fix q<0, p>4, and 1<α<min{p/4,3/2}.  
  
  Let ω_0∈ (0,12 q^2) in the defocusing case and ω_0∈(12q^2,Ω) in the focusing case.  Suppose that the spectral condition in Definition <ref> holds in a neighborhood of ω_0.
  
  For =(q,ω_0,p)>0 and η=η()>0 sufficiently small, we have that if
  
      u_0 = Q_ω_0+ṽ_0withṽ_0_H^1 + ⟨ x⟩^αṽ_0_L^2 <η,
  
  then the corresponding global-in-time solution u to (<ref>) admits a decomposition of the form
  
      u(t) = e^iΦ(t){Q_ω(t)+v(t)}
  
  obeying the following properties:
  
  The phase has the form
  
      Φ(t)=θ(t)+∫_0^t ω(s) ds,
  
  and the modulation parameters obey
  
      |θ̇(t)|+|ω̇(t)| ≲^2 ⟨ t⟩^-2αfor allt≥ 0,
  
  where here and below the implicit constants depend on p and ω_0.  In particular, ω(t) converges to some ω_+ as t→∞, with |ω_0-ω_+|≲^2.
  
  The perturbation v(t) obeys the orthogonality conditions
  
      ⟨ v(t),Q_ω(t)⟩ = ⟨ v(t),i∂_ω Q_ω(t)⟩ = 0for allt≥ 0,
  
  along with the estimates
  
      v(t)_H^1≲,  v(t)_L^r≲⟨ t⟩^-[1/2-1/r],and⟨ x⟩^-αv(t)_L^2≲⟨ t⟩^-α
  
  for all t≥ 0, where r=r(α,p) is a sufficiently large (but finite) exponent.  Moreover, scattering holds in the sense that 
  
      lim_t→∞v(t) - e^tŁ(ω_+)v_+_L^2 = 0
  
  for some v_+∈ L^2.
  
  
   The notation ∂_ω Q_ω(t) appearing in (<ref>) is short-hand for the expression
  
      ∂_ω Q_ω|_ω=ω(t).
   
  The shorter notation appearing in (<ref>) will be used throughout the paper. 
  
  
  Theorem <ref> is conditional on the assumption that ω_0 satisfies the spectral condition in Definition <ref>.  In this paper, we also investigate the range of frequencies for which the spectral condition holds.  Combining some analytic and numerical results in Section <ref>, we arrive at the following conjecture.  Recall that Ω=Ω(q,p) is the minimal unstable frequency introduced above (see also Section <ref> below).
  
   Fix q<0 and p>4. 
   
  
    * In the focusing case (σ=-1), there exists ω_1=ω_1(q,p)∈(12q^2,Ω) such that the spectral condition holds for ω∈(12 q^2,ω_1).
  
    * In the defocusing case (σ=1), the spectral condition holds for all ω∈(0,12q^2).
   
   
  In the focusing case, the frequency ω_1 is identified by the first appearance of a resonance at the edge of the continuous spectrum of ℒ(ω).  By using a shooting method in ω, we can numerically estimate the value of ω_1.  This frequency is always smaller than Ω, and hence Theorem <ref> does not treat the entire range of stable solitary waves in the focusing case.  On the other hand, by computing the mass of Q_ω_1 (representing the largest solitary wave mass addressed by Theorem <ref> in the focusing case), we can see that our result covers a range of solitary waves beyond the small data regime.  Fixing q=-1, we display some computed values of ω_1=ω_1(p) and the mass M of Q_ω_1 in the following table (see also Figure <ref> on page special-omegas1). 
  
  
  
  As p→∞, the minimal unstable frequency Ω itself converges to 12q^2 (see e.g. (<ref>) below).  Consequently, the entire stable regime is restricted to the setting of small solitary waves, and hence Theorem <ref> necessarily collapses to the small data regime as p→∞ in the focusing case.  In the defocusing case, Conjecture <ref> and Theorem <ref> yield asymptotic stability for the entire family of solitary waves. 
  
  In this paper, we do not give a rigorous proof of Conjecture <ref>.  Instead, we focus our attention on the problems of (i) establishing asymptotic stability under the assumed spectral condition and (ii) providing an estimate for the range of ω for which this condition holds.  We carry out (ii) in Section <ref>.  In that section, we also state some further conjectures about the global spectral picture for the operators Ł(ω) (see Sections <ref> and <ref>).  The rest of the paper is then focused on establishing (i).  A more complete analysis of the spectral properties of the operators Ł(ω), as well the extension of Theorem <ref> to handle different spectral conditions in the focusing case, will be addressed in future works.
  
  Theorem <ref> extends our previous work <cit.> beyond the regime of small solitary waves.  In <cit.> we worked with initial frequencies ω_0 very close to 12q^2, in which case the solitary waves look like small multiples of the eigenfunction of the underlying linear problem. In that work, we were able to take the initial perturbation to be small merely in H^1 (i.e. without any weighted assumption), and we proved global space-time bounds and scattering for the corresponding perturbation v.  In this small-data regime, we could view the evolution equation for v as a perturbation of the linear Schrödinger equation with delta potential.  In particular, we could include terms of the form 𝒪(vQ_ω^p) together with the nonlinear terms when proving bootstrap estimates for v.  Consequently, we could rely purely on Strichartz-type estimates for the underlying linear equation, which ultimately allowed us to close the argument without introducing any weighted spaces.  
  
  
  
  In the present setting, the solitary waves Q_ω are no longer assumed to be small, and we must include the 𝒪(vQ_ω^p) terms in the linear part of the equation for v.  In particular, a central component of our argument is the proof of dispersive estimates for the matrix operators e^tŁ(ω).  To close the boostrap argument for v and the modulation parameters (θ,ω), we then rely on the fact that the equations are at least quadratic in v.  For the modulation parameters, this is a consequence of the choice of orthogonality conditions (similar to <cit.>).  As for the equation for v, one has terms that are quadratic or higher in v, as well as some terms involving the derivatives of the modulation parameters.  However, after a change of variables used to remove the time-dependence from the operator in the underlying linear equation, we end up with additional terms that are linear in v (see Proposition <ref>).  While these terms come with additional sources of smallness (via the modulation parameters), their time-decay is no better than that of v itself.  As a result, we can close the bootstrap estimate only if v itself obeys some integrable-in-time decay estimate.  In particular, we must prove estimates that exhibit stronger decay than the usual |t|^-1/2 decay rate associated to Schrödinger-type dispersive estimates in one dimension.  Utilizing such dispersive estimates requires that the initial perturbation belongs to weighted spaces, rather than merely H^1.  
  
  We are able to obtain weighted dispersive estimates in the spirit of <cit.> that exhibit a decay rate of |t|^-3/2. This type of dispersive estimate requires the non-resonance condition appearing in Definition <ref>.  Indeed, such estimates fail for the free Schrödinger equation due to the resonance at zero energy.  In practice, we must use suitably interpolated versions of the weighted dispersive estimates in order to treat the entire mass-supercritical regime p>4.  Indeed, there is a direct relationship between the rate of decay and the number of weights used.  When estimating a weighted norm of the purely nonlinear term |v|^p v, one must balance the decay exhibited by v itself with the growth exhibited by terms like xv_L^2.  As a result, the situation becomes increasingly subtle as the value of p decreases. In fact, although we expect that analogues of Theorem <ref> should hold in the mass-critical and mass-subcritical regime, our approach ultimately breaks down precisely at p=4 (see e.g. Lemma <ref>). 
  
  
  
  Theorem <ref> also fits in the context of the many recent works on asymptotic stability for solitary wave solutions to dispersive equations, for which there is now a huge literature.  For a sample of related results, we refer the interested reader to <cit.>.  Concerning asymptotic stability for the specific model of the 1d NLS with a delta potential, our previous work <cit.> addressed the case of small solitary waves, while Deift–Park <cit.> gave a fairly exhaustive treatment of the case of a cubic nonlinearity and even data (see also <cit.> for some related results).  In the setting of <cit.>, the equation becomes completely integrable and hence is amenable to special techniques such as inverse scattering and nonlinear steepest descent.  Another related work is that of <cit.>, which utilized a virial-type argument to prove global space-time estimates in negative exponentially weighted spaces for the perturbation around small solitary waves for (<ref>).  Finally, <cit.> showed the failure of (unmodified) scattering to solitary waves for (<ref>) with long-range nonlinearities (i.e. p≤ 2).
  
  In terms of the techniques employed here, the works that are most closely-related are those of Buslaev–Perelman <cit.>, Gang–Sigal <cit.>, and Krieger–Schlag <cit.>.  The works <cit.> concerned asymptotic stability for 1d Schrödinger equations (with or without potential) with suitable nonlinearities, while <cit.> addressed the problem of stable manifolds for the 1d NLS with mass-supercritical power-type nonlinearities.  A common theme in all of these works, as well as the present work, is the development of suitable dispersive estimates for the (matrix) operators arising from the linearization of the nonlinear equation around the moving solitary wave, coupled with a bootstrap argument to derive decay estimates for the perturbation and the modulation parameters.  
  
  The rest of this paper is organized as follows:
  
  
    * In Section <ref>, we first collect some notation.  We formally introduce the Schrödinger operator H=-12∂_x^2 + qδ_0 in Section <ref>, and in Section <ref> we provide a more detailed introduction to the solitary wave solutions Q_ω.  
  
    * Section <ref> contains the spectral analysis for the operator Ł(ω).  In particular, we provide some numerical evidence for the claims made in Conjecture <ref>.  We also give some further conjectures related to the global spectral picture for the operators Ł(ω). 
  
    * In Section <ref>, we prove dispersive estimates for the evolution group e^tŁ(ω) under the assumption that ω obeys the spectral condition Definition <ref>.  The argument requires the construction of some particular solutions to an associated eigenvalue problem, which we carry out in Appendix <ref>.  
  
    * Finally, in Section <ref>, we prove the main result, Theorem <ref>. 
  
  
  
  
   §.§ Acknowledgements
   S.M. was supported by JSPS KAKENHI Grant Numbers JP17K14219, JP17H02854, JP17H02851, and JP18KK0386. J.M. was supported in part by a Simons Collaboration Grant. J.S was supported by JSPS KAKENHI Grant Numbers JP17H02851, JP19H05597, and JP20H00118.  We are grateful to Rowan Killip and Maciej Zworski for helpful suggestions concerning numerical approaches for the spectral analysis. 
  
  
  
  
  
  
  
  
  
  
  
  
  § NOTATION AND PRELIMINARIES
  
  
  We write A≲ B to denote the inequality A≤ CB for some constant C>0.  We denote the dependence of the implicit constants on various parameters via subscripts, e.g. A≲_p B denotes A≤ CB for some C=C(p)>0.  We will also make use of the big-oh notation 𝒪, again indicating dependence on various parameters via subscripts.  We use the Japanese bracket notation ⟨ x⟩=√(1+x^2). We denote the transpose of a matrix or vector using the notation ^t. 
  
  We utilize the standard Lebesgue and Sobolev spaces.  In addition, we will make use of Lorentz spaces, defined via the quasi-norms
  
      f_L^a,b = λ| {x:|f(x)>λ}|^1/a_L^b((0,∞),dλ/λ)
  
  for 1≤ a<∞ and 1≤ b ≤∞.  In particular, we have L^a,a=L^a, while L^a,∞ corresponds to weak L^a.  In general, one has the embedding L^a,b↪ L^a,b' for b<b'.  Lorentz spaces are convenient in our setting essentially due to the fact that pure powers belong to to Lorentz spaces, namely, |x|^-a∈ L^d/a,∞(^d) for 0<a≤ d. 
  
  Many standard inequalities have Lorentz-space analogues (see e.g. <cit.>).  For example, we will make use of Hölder's inequality in the form
  
      fg_L^a,b≲f_L^a_1,b_1g_L^a_2,b_2
  
  for 1≤ a,a_1,a_2<∞ and 1≤ q,q_1,q_2≤∞ satisfying 1a=1a_1+1a_2 and 1b=1b_1+1b_2. 
  
  We use the following notation for inner product:
  
      ⟨ f,g⟩ = ∫f̅(x) g(x) dx.
  
  We will regularly identify a complex-valued function f=f_1+if_2 with the vector [f_1 f_2]^t taking values in ^2.  Our definition of inner product corresponds to the standard inner product on ^2-valued functions in the sense that
   
      ⟨[[ f_1; f_2 ]],[[ g_1; g_2 ]]⟩ =  ∫ f_1(x)g_1(x) dx + ∫ f_2(x)g_2(x) dx = ⟨ f,g⟩.
  
  
  Finally, we recall the following standard Pauli matrices: 
  
      σ_1=[[ 0 1; 1 0 ]],   σ_3 = [[   1   0;   0 - 1 ]].
  
  
  
  
  
  
  
   §.§ The linear Schrödinger equation with a delta potential
  
  
  The linear Schrödinger equation with a delta potential is a classical model in quantum mechanics, which is discussed extensively in <cit.>.  We consider the case of an attractive potential, that is, we let
  
      H=-12∂_x^2 + qδ(x) with q<0.
  
  More precisely, the operator H is defined by H=-12∂_x^2 on its domain, given by
  
      D(H)={u∈ H^1()∩ H^2(\{0}):∂_x u(0+)-∂_x u(0-)=2qu(0)},
  
  where 0± denotes limit from the right/left, and H extends to a self-adjoint operator on L^2 with purely absolutely continuous spectrum equal to [0,∞).  We frequently refer to the condition appearing in D(H) as the jump condition at x=0.  If q>0 (the repulsive case), then H has no eigenvalues, while if q<0 (the attractive case, under consideration in this paper) then H has a single negative eigenvalue -12q^2 with a one-dimensional eigenspace spanned by  ϕ_0(x):=|q|^1/2 e^-|q| |x|.
  
  
  
  
  
   §.§ Solitary wave solutions
    Equation (<ref>) admits a family of solitary wave solutions to (<ref>) of the form u(t,x)=e^iω tQ_ω(x), where Q_ω solves (<ref>). In the defocusing case (i.e. with f(u)=+|u|^p u), the frequency ω takes values in (0,12q^2) and the solitary waves take the explicit form
  
      Q_ω(x) = [(p+2)ω2]^1/psinh^-2/p[ p√(ω2)|x|+arctanh(√(2ω)|q|)].
    
  For ω=0, one also obtains the solution
  
      Q_0(x) = [(p+2)|q|^2(p|q| |x|+2)^2]^1/p,
  
  which belongs to L^2 provided p<4.  As we are restricting our attention to p>4, we will not directly consider this case in the present work. 
  
  In the focusing case (i.e. with f(u)=-|u|^p u), the frequency ω takes values in (12q^2,∞) and the solitary waves take the explicit form
  
      Q_ω(x) = [(p+2)ω2]^1/pcosh^-2/p[ p√(ω2)|x|+arctanh(|q|√(2ω))].
  
  To derive these explicit formulas, one views (<ref>) as a one-dimensional ODE on each side of x=0 coupled with the jump condition appearing in (<ref>).  See e.g. <cit.>.  
  
  The stability properties of these solitary waves were studied in <cit.>. In the defocusing case, all solitary waves are orbitally stable <cit.>. In the focusing case, the authors of <cit.> utilized the general theory of <cit.> to prove the existence of Ω such that Q_ω is stable for ω∈(12q^2,Ω) and unstable for ω∈(Ω,∞).  Instability for ω=Ω was subsequently established in <cit.>.  Here Ω is
  defined as the solution to 
  
      p-4p∫_arctanh(|q|/√(2ω))^4/p(x) dx = |q|√(2ω)[1-q^22ω]^2/p-1
  
  and represents the unique frequency at which ddωQ_ω_L^2^2=0. In fact, we have
  
      ddω12Q_ω_L^2^2 = ⟨ Q_ω,∂_ω Q_ω⟩ >0for12q^2 <ω<Ω
  
  in the focusing case, while ⟨ Q_ω,∂_ω Q_ω⟩<0 for all ω∈(0,12q^2) in the defocusing case.  In particular, this inner product will be nonzero for all ω under consideration in our main result (Theorem <ref>).
  
  We also note that by using the explicit formulas for Q_ω, we have that ⟨∂_ω⟩^2 Q_ω∈ H^1,k for any k≥ 0.  Moreover, the bounds are uniform over compact sets of ω. 
  
  
  
  
  
  
  
  
  
  
  § SPECTRAL ANALYSIS
  
  
  In this section we discuss the spectral properties of the family of operators Ł(ω) arising from the linearization of (<ref>) around the nonlinear bounds states Q_ω.  We recall
  
      Ł=Ł(ω) := [[    0  H+ω; -H-ω    0 ]]+σ Q_ω^p[[      0      1; -(p+1)      0 ]],
  
  where H=-12∂_x^2 + qδ_0 with q<0 (see Section <ref>) and Q_ω are the nonlinear bound states (see Section <ref>).  In particular, we recall that in the defocusing case σ=+1, we have ω∈(0,12q^2), while in the focusing case σ=-1 we have ω∈(12q^2,∞).  We also recall that ⟨ Q_ω,∂_ω Q_ω⟩=12ddωQ_ω_L^2^2 is non-zero except for the case ω=Ω in the focusing case, which marks the transition between stable/unstable solitary waves (see <cit.> and Section <ref>). 
  
  We view Ł(ω) as acting on ^2-valued functions, where we identify a complex-valued function f=f_1+if_2 with the vector [f_1 f_2]^t.  We introduce the notation
  
      L_- = H+ω+σ Q_ω^p,    L_+ = H+ω+σ(p+1)Q_ω^p,
  
  so that
  
      Ł=Ł(ω)=[[    0  L_-; -L_+    0 ]].
  
  
  We will present a blend of analytic results, numerical results, and conjectures related to the spectral properties of Ł(ω).  Our main goals in this section are are (i) to give a qualitative description of the global spectral properties of Ł(ω) and (ii) for fixed values of q and p, estimate the range of frequencies ω for which the spectral condition appearing in Definition <ref> holds.  By addressing item (ii), we can verify that our main result (Theorem <ref>) extends beyond the regime of small solitary waves, which is the primary ambition of this paper.  Indeed, we need only compute the masses corresponding to the range of admissible frequencies.  
  
  The results in this section provide some evidence for Conjecture <ref> stated in the introduction.  In addition, we will state some further conjectures concerning the spectral properties of ℒ(ω).  We plan to undertake a more complete analysis in future work. 
  
  We begin by collecting some preliminary results concerning the spectrum of Ł(ω).  From (<ref>) we see that the continuous spectrum σ_c(ℒ(ω)) of Ł(ω) equals i[ω,∞)∪ i[-ω,-∞) for all ω.
  
  Next, using some known facts about the operators L_± (see e.g. <cit.>), we can establish the following results concerning the generalized kernel of Ł.  In particular, we have a two-dimensional generalized kernel for almost every value of ω.  The only exception is ω=Ω in the focusing case, at which point this subspace becomes four-dimensional.  We refer the reader to <cit.> for a discussion related to taking powers of Ł.
  
   Let ω∈(0,12q^2) in the defocusing case, or ω∈(12q^2,∞)\{Ω} in the focusing case, so that
  
      ⟨ Q_ω,∂_ω Q_ω⟩≠ 0.
  
  Then
  
      Ł^k = span{∂_ω Q_ω, i Q_ω}for allk≥ 2.
  
  
  In the focusing case, if ω=Ω, then there exist h and g such that 
  
      Ł^k = span{∂_ω Q_ω,iQ_ω,h,ig}fork≥ 4.
  
  
  
  
   Throughout the proof, we use c,c' to denote constants that may change from line to line.
  
  We first suppose ω≠Ω. We will rely on the following facts about the operators L_± (see e.g. <cit.>):
  
  
    (i) L_-f = 0  f =c Q_ω.
  
    (ii) L_+f = 0  f = 0.
  
    (iii) L_+f = cQ_ω f=c'∂_ω Q_ω.
  
    (iv) L_-f = c∂_ω Q_ω L_-f = 0  f=c'Q_ω.
  
  Indeed, (i) is just the fact that the kernel of L_- is one-dimensional and spanned by Q_ω, while (ii) is the fact that L_+ has no kernel.  Similarly, (iii) follows from L_+∂_ω Q_ω = -Q_ω and (ii). Finally, the first implication in (iv) follows from the fact that ∂_ω Q_ω∉[(L_-)]^⊥, which follows from (<ref>) and (i). 
  
  We proceed by induction.  First, for k=2, we write
  
      Ł^2=[[ - L_-L_+        0;        0 -L_+ L_- ]].
  
  Thus if Ł^2 f = 0, we have L_-L_+ f_1 = 0, yielding L_+ f_1 = cQ_ω and so f_1 = c'∂_ω Q_ω.  Similarly, L_+L_-f_2 = 0, yielding L_- f_2=0 and hence f_2=cQ_ω. 
  
  Next, we suppose the result holds for some k and suppose that
  
      Ł^k+1 f = Ł^k[[  L_- f_2; -L_+ f_1 ]] = 0.
  
  Then, by the inductive hypothesis, we must have L_- f_2 = c∂_ω Q_ω, yielding f_2 = c'Q_ω.  Similarly, L_+ f_1 = cQ_ω, yielding f_1 = c'∂_ω Q_ω.  This completes the proof in the case ω≠Ω.
  
  If ω=Ω, we have ⟨ Q_ω,∂_ω Q_ω⟩ = 0, and thus we may find g such that L_-g = ∂_ω Q_ω.  We then define h=L_+^-1g, so that L_+h = g.  To prove the result we can then carry out an induction argument as above.
  
  The eigenvalues of Ł(ω) must be either real or purely imaginary.  Formally, this follows from the fact that Ł^2 is self-adjoint (see (<ref>)), so that any eigenvalue λ of Ł must obey λ^2∈.  To be more precise, we will use an argument from <cit.>. (See also <cit.>.)
  
   Eigenvalues of Ł must belong to ∪ i.  Furthermore, nonzero eigenvalues must appear in ± or conjugate pairs. 
   
  
   Suppose Ł[u v]^t = λ[u v]^t for some λ≠ 0 and nontrivial [u v]^t.  Then
  
      L_+ u =-λ v andL_-v = λ u,
  
  which implies
  
      ⟨ u,L_+ u⟩ = -λ∫u̅ v dx and⟨ v,L_-v⟩ = λ∫v̅ u dx = λ∫u̅ v dx,
  
  so that
  
      ⟨ u,L_+u⟩⟨ v,L_-v⟩ = -λ^2|∫u̅ v dx|^2.
  
  
  Now, if v=0, then since L_+ has no kernel we also find u=0, yielding a contradiction.  Similarly, since λ≠ 0, if u=0 then v = λ^-1L_+ u=0 and we obtain a contradiction. In particular, L_- v = λ u≠ 0 and hence v is not a multiple of Q_ω.  This implies ⟨ v,L_-v⟩>0, which (using the identities above and recalling λ≠ 0) implies ∫u̅ v dx ≠ 0.  In particular, we deduce
  
      λ^2 = -⟨ u,L_+ u⟩⟨ v,L_-v⟩/|∫u̅ v dx|^2∈,
  
  yielding λ∈∪ i.  
  
  The fact that eigenvalues occur in pairs follows from symmetry properties of Ł (see e.g. <cit.>). 
  
  
  
  
   §.§ The focusing case.
   In the focusing case, we have ω∈ (12q^2,∞), with orbital stability for ω∈ (12q^2,Ω) and instability for ω∈[Ω,∞).  We can try to piece together the global picture of the spectrum by first considering three regimes, which we can then try to connect in a consistent way.  In particular, we consider: (i) ω→12 q^2, (ii) ω near Ω, and (iii) ω→∞. 
   
  (i) First, from the explicit form of the solitary wave solutions (see Section <ref>), the potentials in L_± have the  form
  
      c_±·ωcosh^-2[p√(ω2)|x|+arctanh(|q|√(2ω))],
  
  where c_-=-p+22 and c_+=-(p+1)(p+2)2. In particular, both potentials tend to zero as ω→12 q^2, and so formally the operator Ł(ω) converges to 
  
      Ł(12q^2):=[[            0     H+12 q^2; -[H+ 12 q^2]            0 ]].
  
  Now, the operator H=-12∂_x^2+qδ_0 has a single negative eigenvalue 12 q^2 with a one-dimensional eigenspace spanned by ϕ_0(x) = |q|^1/2e^-|q| |x|; otherwise, H has continuous spectrum in [0,∞).  Thus Ł(12 q^2) has the two-dimensional kernel spanned by {ϕ_0,iϕ_0}, and we can derive that any other eigenvalues must come in purely imaginary pairs.  However, the existence of a purely imaginary eigenvalue would imply the existence of a negative eigenvalue for the corresponding operator (̋12q^2) (see (<ref>)), which in this case is guaranteed to be nonnegative.  Therefore, we can see that our desired spectral condition holds formally as we send ω→12 q^2, and so we expect the condition to hold at least in a neighborhood of 12q^2. (See also Section <ref> below.)
  
  (ii)  Analysis of the type carried out in Comech–Pelinovsky <cit.>, together with the stability/instability results of <cit.>, can provide a picture of the spectrum in a small neighborhood of ω=Ω.  In particular, the `extra' two dimensions of the generalized kernel at ω=Ω (see Lemma <ref>) will split off into a pair of eigenvalues, which will be real in the unstable case (ω>Ω) and purely imaginary in the stable case (ω<Ω). 
  
  (iii) To study the regime ω→∞, we introduce the scaling
  
      S_ω f (x) := f(√(ω)x).
  
  
  A direct computation using (<ref>) shows that
  
      L_±(ω)     = ω S_ω[-12∂_x^2 + 1 + c_±cosh^-2(p|x|√(2)+arctanh(|q|√(2ω)))+q√(ω)δ_0]S_ω^-1
          = ω S_ω[L_±^0 + E_q,ω,±]S_ω^-1,
  
  where L_±^0 denote the operators arising in the linearization around the ground state for the free NLS and E_q,ω,± will be `perturbative' terms.  More precisely, we recall that
  
      R(x) = (p+22)^1/pcosh^-2/p(px√(2))
  
  is the ground state soliton for the free NLS
  
      i∂_t u + 12∂_x^2 u = -|u|^p u,
  
  and that we have the corresponding linearized operators 
  
      L_+^0 = -12∂_x^2 + 1 - (p+1) R^p,   L_-^0 = - 12 ∂_x^2 + 1 - R^p.
  
  The potentials E_q,ω,± introduced above are therefore given explicitly by 
  
      E_q,ω,± = c_±[cosh^-2(p|x|√(2)+arctanh(|q|√(2ω))) - cosh^-2(p|x|√(2))]+q√(ω)δ_0.
  
  The operator then takes the form
  
      Ł(ω) = ω S_ω[ Ł^0 + ℰ_q,ω] S_ω^-1,
  
  where Ł^0 is built out of L_±^0 in the usual way and ℰ_q,ω tends to zero as ω→∞.  In particular, we expect that in the limit as ω→∞, the spectrum of Ł(ω) will be related to that of Ł^0.  The spectral properties of this latter operator have been studied in great detail in previous works (see e.g. <cit.> or <cit.>). In the mass-supercritical setting, the discrete spectrum of Ł^0 is six-dimensional, with a four-dimensional generalized kernel and a pair of real eigenvalues. 
  
  By connecting the three regimes described above, we can arrive at a conjecture for a qualitative description of the spectrum of Ł(ω) for ω∈(12q^2,∞).
  
   
  
  There exists ω_1∈(12q^2,Ω) such that Ł(ω_1) has a pair of resonances at ± iω_1. For ω∈(12q^2,ω_1) the spectral condition of Definition <ref> holds.  For ω>ω_1, there exists a first eigenvalue pair ±λ_1(ω).  This pair corresponds to the pair of eigenvalues described in regime (ii) above.  In particular, we have 
  
      ±λ_1(ω) ∈ i\σ_c(ℒ(ω))   forω_1<ω<Ω
      {0}   forω=Ω
         forω>Ω.
  
  
  There exists ω_2>ω_1 such that Ł(ω_2) has a second pair of resonances at ± iω_2.  For ω>ω_2, there is a second eigenvalue pair ±λ_2(ω). This pair corresponds to two of the dimensions of the generalized kernel of Ł^0 in regime (iii) above (see (<ref>)).  In particular, we have have 
  
      ±λ_2(ω) ∈ i\σ_c(Ł(ω))forω>ω_2,
  
  with ±λ_2(ω)→ 0 as ω→∞.
  
  For all ω∈(12q^2,∞), there are no eigenvalues embedded in the continuous spectrum of Ł(ω). 
  
  
  The claims made in Conjecture <ref> are depicted in the following figures, which give qualitative representations of the spectrum of Ł(ω) as ω varies in (12q^2,∞).  We have presented the scenario in which the first pair of eigenvalues crosses the origin before the emergence of the second eigenvalue pair (which we found to be the case for p>p_0≈ 4.54).  
  
  
  
      < g r a p h i c s >
  
      < g r a p h i c s >
  
  
  
  
      < g r a p h i c s >
  
      < g r a p h i c s >
  
  
  
  
      < g r a p h i c s >
  
      < g r a p h i c s >
  
  
  
  
      < g r a p h i c s >
  
  
  
  
  
    §.§.§ Numerical search for resonances
   To provide evidence for the conjecture above, we undertook a numerical search for resonances of Ł(ω) appearing at ± iω.  For all numerical investigations, we fixed the parameter q=-1 and focused on finding the resonance at iω.  
  
  To formulate the problem, we first rewrite the resonance equation
  
      Ł(ω)[[ f; g ]] = iω[[ f; g ]]
    
  for complex-valued f=f_1+if_2 and g = g_1 + ig_2 as a system of four second-order ODEs, namely:
  
      L_-g_1 = -ω f_2,    L_- g_2 = ω f_1, 
          L_+f_1 = ω g_2,    L_+ f_2 = -ω g_1.
  
  As usual, to deal with the presence of the delta potential we view each equation as an ODE on _-∪_+ together with the jump condition at x=0 (see (<ref>) at x=0). On each of _±, the operator H acts as -12∂_x^2. 
  
  As these equations are linear, it suffices to look for even or odd solutions separately.  In each case, it is enough to solve the ODE on _+, impose a suitable boundary condition at x=0, and extend the solution to  by taking either the even or odd reflection.  For even solutions, we obtain the boundary condition
  
      u'(0) = qu(0),
  
  whereas for odd solutions we obtain 
  
      u(0)=0.
  
  
  To solve (<ref>) numerically, we utilize the bvp4c Matlab package, viewing the equations as boundary-value problems on [0,x_0] for sufficiently large x_0.  We chose x_0=50 and found that increasing x_0 further did not change the numerical results.  The ODE system (<ref>) is straightforward to study numerically, as the potentials appearing in L_± are given explicitly using the formulas for the nonlinear bound states given in Section <ref>. 
  
  We impose either (<ref>) or (<ref>) for each of f_1, f_2, g_1, and g_2, depending on whether we seek even or odd solutions.  For the remaining four boundary conditions, one possibility is to impose u'(x_0)=0 for each of f_1, f_2, g_1, and g_2.  This is consistent with the existence of a resonance, which should correspond to a solution in (L^∞∩Ḣ^1)\ L^2.  It would also be consistent with the existence of an eigenvalue. However, this choice of boundary condition turns out to be difficult to work with numerically, as it tends to yield solutions so small that they cannot be meaningfully distinguished from the zero solution.
  
  Thus, for the remaining four boundary conditions we proceed as follows.  Recalling that we are looking for a resonance, we first seek to impose the conditions f_1→ 1 and f_1'→ 0 as x→∞. Thus, for the numerical computations we impose 
  
      f_1(x_0)=1and f'(x_0)=0.
  
  Next, by elliptic regularity it is natural to assume that the second derivatives vanish as x→∞ for solutions in L^∞∩Ḣ^1.  In this case, taking the limit as x→∞ in (<ref>) yields the two final numerical boundary conditions 
  
      f_1(x_0)=g_1(x_0) and f_2(x_0)=-g_2(x_0).
   
  
  We now employ a shooting method with the parameter ω.  That is, we fix a discrete set D and numerically solve the system (<ref>) with the boundary conditions above for each ω∈ D.  Most solutions have asymptotically linear behavior as x→ x_0, corresponding to solutions to (<ref>) that grow in magnitude as x→∞.  On the other hand, a resonance should be asymptotically flat.  To look for possible resonances, we therefore measure the norm of the derivatives f_1', f_2', g_1', and g_2' in a neighborhood of x_0.  If these norms become small around some ω'∈ D, it suggests the existence of a resonance near ω'.  We may then take a refined set D of frequencies near ω'  and start the search again.  After several iterations, we can identify the resonance.  The idea of the method is illustrated briefly in Figure <ref>.
  
   
  
  For a range of values of p≥ 4, we consistently found one even resonance and one odd resonance.  The even resonance always occurred at a value ω_1<Ω, which is what we expected in light of the discussion above.  As p increases, the odd resonance initially occurs at a value ω_2<Ω and later occurs at ω_2>Ω, with the transition occurring around the value p_0≈ 4.54.  In particular, for p<p_0 and ω_2<ω<Ω, two pairs of imaginary eigenvalues can co-exist, whereas for p>p_0 this cannot occur.  The shapes of the resonances appear already in Figure <ref>.  Closer views of these solutions near the origin are displayed in Figure <ref>.
  
  
  
  In Figures <ref> and <ref>, we display our main numerical results on the existence of resonances for Ł(ω).  Here q=-1 is fixed throughout.  In Figure <ref>, we plot the value of the even resonance ω_1 and the threshold frequency Ω against the power p.  We also display the mass of Q_ω_1 as a function of ω_1, representing the upper bound on solitary wave mass for which our main result (Theorem <ref>) applies in the focusing case.  In particular, this demonstrates that we can treat solitary waves that are 𝒪(1) in L^2, and hence our result does indeed extend beyond the small solitary wave regime.  We list some  computed values of ω_1 and M(Q_ω_1) in the table on page the-table.  As p→∞, one finds that Ω (and hence ω_1) converges to 12q^2, so that one has stability of any type only for small solitary waves.  On the other hand, as p decreases to 4 we obtain Ω→∞, corresponding to the fact that all solitary waves are stable when p=4, while we are still able to find an even resonance appearing at ω_1≈ 2.6648.
  
  
  
  
  In Figure <ref>, we also plot  the odd resonance ω_2, which appears to satisfy ω_2→∞ as p→∞.  As these resonances always appear after the even resonance, they are not directly relevant for the analysis of the present paper.  
  
  
   
  
  
    §.§.§ Embedded eigenvalues.
   In Conjecture <ref>, we assert that there are no eigenvalues embedded in the continuous spectrum of Ł(ω).  This is a generic assumption in stability problems that has been established rigorously in some related settings (see e.g. <cit.>); see also <cit.> for a numerical approach.  Some of the theory that will be developed in the next section already rules out the possibility of embedded eigenvalues of sufficiently large magnitude (in both the focusing and defocusing cases).  In that section, the analysis is done in terms of the operator
  
      =̋[[  H+ω    0;    0 -H-ω ]] + σ Q_ω^p [[  p+22    p2;   -p2 -p+22 ]],
  
  which is connected to ℒ via 
  
      -iU^∗ℒU=,̋   U=1/√(2)[[  1  1;  i -i ]].
  
  Ruling out imaginary eigenvalues of Ł is then equivalent to ruling out real eigenvalues of $̋.  In Proposition <ref>, we define a matrix-valued functionD(ξ)(given as the matrix Wronskian of two particular solutions to the generalized eigenvalue problem for$̋) and show that ω+12ξ^2 is an eigenvalue of $̋ if and only ifD(ξ)=0.  On the other hand, in Lemma <ref> (see (<ref>)) we derive that 
  
      D(ξ) = -4iξ√(ξ^2+4ω) + 𝒪(1) as|ξ|→∞.
  
  This precludes the existence of large eigenvalues forH, and so rules out the possibility of large imaginary eigenvalues forŁ.
  
  We can consider the limiting caseω→12q^2.  In this case, one can compute explicitly to derive
  
      D(ξ) → -4[iξ-q][√(ξ^2+4ω)+q],
  
  which is always nonzero.  Using continuity with respect toω, this implies the absence of embedded eigenvalues at least forωin a neighborhood of12q^2.  In the present paper, we take the absence of any embedded eigenvalues forŁ(ω)as an assumption (see the spectral condition in Definition <ref>).  We will address this problem in more detail in future work. 
  
  
  
  
  
   §.§ The defocusing case.
   In the defocusing case, orbital stability holds for allω∈(0,12q^2).  Our conjecture for the defocusing case is that the spectral condition in Definition <ref> holds for allω∈(0,12q^2), so that Theorem <ref> applies to the entire family of solitary wave solutions.  In what follows, we give some evidence for this conjecture.  A more rigorous analysis will be the topic of future work. 
  
  The limitω→12 q^2is essentially the same as in the focusing case, as the solitary wave solutions converge to zero.  Thus we once again expect the spectral condition to hold at least in a neighborhood of12q^2.  As in the focusing case, we carried out a numerical search for resonances at the edge of the continuous spectrum signaling the emergence of imaginary eigenvalues.  In contrast to the focusing case, we found none.  In addition, orbital stability suggests that real eigenvalues should not occur, as these would correspond to exponentially growing modes for the underlying linearized equation. This leads us to the assertion that for allω∈(0,12q^2), the discrete spectrum ofŁ(ω)consists only of the2dgeneralized kernel. Similar to the focusing case, we also assert that embedded eigenvalues do not occur, with the special cases of large magnitude eigenvalues already precluded as in Section <ref>.  
  
  Proceeding analogously to the focusing case, we may also attempt to connect the regimeω→12q^2to the other limiting regimeω→0.  In this latter regime, we obtain the operator
  
      Ł(0) := [[  0  H; -H  0 ]] + V(x)[[      0      1; -(p+1)      0 ]],
  
  where
  
      V(x) = (p+2)|q|^2/(p|q||x|+2)^2
  
  (see (<ref>)).  If we additionally sendq→0in (<ref>), then we haveV→0andH→-12∂_x^2, and the discrete spectrum consists only of a resonance of multiplicity 2 at the origin (corresponding to the constant functions[1 0]^tand[0 1]^t).  Forq<0, we retain this resonance at zero, which corresponds to the limit of the2dgeneralized generalized kernel ofℒ(ω).  We obtain a resonance (rather than an eigenvalue) due to the fact that in the mass-supercritical regime, the limits ofQ_ωand∂_ωQ_ωasω→0fail to belong toL^2(see (<ref>)).  In light of the above, we expect that there is no additional discrete spectrum for the limiting operatorŁ(0).  Our assertion that the discrete spectrum ofŁ(ω)is given only by its2dgeneralized kernel for allω∈(0,12q^2)then connects these two limiting regimes in a consistent way.
  
  
  
   §.§ Projection formula.
   To conclude this section, we establish a formula for the projection away from the continuous spectrum forŁ(ω), under the assumption thatωobeys the spectral condition appearing in Definition <ref>.  The argument is the same as in <cit.>.  
  
   Assume that ω is such that (<ref>) holds and Ł(ω) has no resonances and no eigenvalues other than zero.  Let P_c denote the projection onto the continuous spectrum of Ł(ω). Then
  
      (I-P_c) f = ⟨ f,Q_ω⟩/⟨ Q_ω,∂_ω Q_ω⟩∂_ω Q_ω + ⟨ f,i∂_ω Q_ω⟩/⟨ Q_ω,∂_ω Q_ω⟩ iQ_ω.
   Using Lemma <ref>, we find that the range of I-P_c equals the span of {∂_ω Q_ω,iQ_ω}. Thus we have
  
      (I-P_c)f = a_1 ∂_ω Q_ω + ia_2Q_ω
  
  for some a_1,a_2∈.   In particular,
  
      ⟨ (I-P_c)f, Q_ω⟩    = a_1⟨ Q_ω,∂_ω Q_ω⟩, 
      ⟨ (I-P_c)f,i∂_ω Q_ω⟩    = a_2 ⟨ Q_ω,∂_ω Q_ω⟩.
  
  As the transpose P_c^t is the projection onto the continuous spectrum of Ł^t, we derive that (I-P_c^t)Q_ω=Q_ω and (I-P_c^t)i∂_ω Q_ω = i∂_ω Q_ω.  Solving for a_1 and a_2 now yields the result. 
  
  
  
  
  
  
  
  
  § DISPERSIVE ESTIMATES
  
  
  In this section we derive dispersive estimates 
  for the evolution groupe^tℒ, whereℒ=ℒ(ω)is as in (<ref>).  We first observe that 
  
      -iU^∗ℒU=,̋
  
  where 
  
      U=1/√(2)[[  1  1;  i -i ]]
  
  and
  
      =̋_̋0+:= [[  H+ω    0;    0 -H-ω ]] + σ Q_ω^p [[  p+22    p2;   -p2 -p+22 ]].
  
  In particular, it is equivalent to prove dispersive estimates for the unitary groupe^it, as we havee^tℒ=Ue^itU^∗.
  
  
   Suppose ⟨ Q_ω,∂_ω Q_ω⟩≠ 0 and the spectral condition in Definition <ref> holds. Then, writing P_c for the projection to the absolutely continuous spectrum of $̋, we have 
  
      e^itP_c φ_L^∞   ≲ |t|^-1/2φ_L^1, 
      ⟨ x⟩^-1e^itP_cφ_L^∞   ≲ |t|^-3/2⟨ x⟩φ_L^1.
  
  In addition, we have theL^2bound
  
      e^itP_cφ_L^2≲φ_L^2.
   Throughout this section the parameter ω is fixed and will often be suppressed in formulas.  In Section <ref>, however, we will need to apply the dispersive estimates for varying choices of ω.  The proof below will demonstrate that the implicit constants appearing in (<ref>) and (<ref>) depend certain norms of the potential 𝒱, and in particular they depend continuously on the parameter ω.
  
  To prove Proposition <ref>, we will adapt the arguments of Goldberg–Schlag <cit.> and Krieger–Schlag <cit.>. The starting point for the argument is the following spectral resolution of the propagator (see <cit.>): 
  
      2π i⟨ e^it P_cφ,ψ⟩
          =   [∫_-∞^-ω+∫_ω^∞]e^itλ⟨{(-̋λ-i0)^-1-(-̋λ+i0)^-1}φ,ψ⟩ dλ
          =    e^itω∫_0^∞ e^itξ^2/2ξ⟨{(-̋12ξ^2-ω+i0)^-1-(-̋12ξ^2-ω-i0)^-1}φ,ψ⟩ dξ
      
            + e^-itω∫_0^∞ e^-itξ^2/2ξ⟨{(+̋12ξ^2+ω+i0)^-1-(+̋12ξ^2+ω-i0)^-1}φ,ψ⟩ dξ,
  
  where we have changed variables viaλ=12ξ^2+ωon(ω,∞)andλ=-12ξ^2-ωon(-∞,-ω). To estimate these integrals, we split into low-energy and high-energy regions, separated by a threshold energyλ=C𝒱_L^1for some sufficiently largeC.  
  
  We will first compute the integral kernel of the resolvent and the spectral measure by using solutions to the associated generalized eigenvalue problem (i.e. by means of the distorted Fourier transform).  Using this representation, we will then estimate the contribution of the low-energy regime to the dispersive estimates (<ref>) and (<ref>).  Finally, we will estimate the contribution of the high-energy regime to these estimates by utilizing instead the Born series expansion for the resolvent.  
  
  
  
  
  
   §.§ Representation formula
  
  In this section, we compute the integral kernel of the resolvent in terms of solutions to the generalized eigenvalue problem.  We follow the general strategy set out in <cit.>.  We consider the generalized eigenvalue problem
  
      f̋ = (12ξ^2 + ω)f
  
  and begin with some basic ODE considerations (see Appendix <ref> for more details).  First, for^2-valued functionsfandg, we define the Wronskian
  
      W[f,g](x) := ^tf'(x)g(x) - ^tf(x) g'(x)∈,
  
  and similarly for matrix-valued solutionsF,Gto (<ref>) we define the Wronskian
  
      𝒲[F,G](x) = ^tF'(x)G(x) - ^tF(x)G'(x)∈^2× 2.
  
  The Wronskian of two solutions to (<ref>) is independent ofx. For the vector-valued case, one shows using (<ref>) thatddxW[f,g]=0forx≠ 0, so that the Wronskian is constant on(-∞,0)and on(0,∞).  The jump condition guarantees the two constants are the same.  For the matrix-valued case, one can then follow the argument of <cit.>. 
  
  We next introduce some particular solutions to (<ref>).  
  
   The problem (<ref>) admits four solutions f_1,f_2,f_3,f̃_4 obeying the following. Defining
  
      μ=√(ξ^2 + 4ω),
  
  there exists γ>0 such that for x≥ 0, ℓ∈{0,1,2,…}, and k∈{0,1}, we have
  
      | ∂_ξ^ℓ∂_x^k{ e^-ixξf_1(x,ξ)-[[ 1; 0 ]] }|    ≲μ^-1⟨ x⟩^ℓ+1e^-min{γ,μ}x, 
      
      | ∂_ξ^ℓ∂_x^k{ e^ixξf_2(x,ξ)-[[ 1; 0 ]] }|    ≲μ^-1⟨ x⟩^ℓ+1e^-min{γ,μ}x, 
      | ∂_ξ^ℓ∂_x^k{ e^μ xf_3(x,ξ)-[[ 0; 1 ]] }|    ≲μ^-1e^-γ x    (ℓ=k=0), 
      μ^-ℓ-1e^-2γ x    (ℓ+k≥1),
      
      | ∂_ξ^ℓ∂_x^k{ e^-μ xf̃_4(x,ξ)-[[ 0; 1 ]] }|    ≲μ^-1⟨ x⟩^2e^-min{γ,μ}x    (ℓ=0,k≥ 0), 
      μ^-2x^2+ℓ e^-min{γ,μ}    (ℓ≥ 1, k≥ 0).
  
  The implicit constants above depend continuously on ω. Furthermore,
  
      f_1(x,-ξ)=f_1(x,ξ)=f_2(x,ξ),   f_2(x,-ξ)=f_2(x,ξ)=f_1(x,ξ), 
         f_3(x,-ξ)=f_3(x,ξ)=f_3(x,ξ),  f̃_4(x,-ξ)=f̃_4(x,ξ)=f̃_4(x,ξ),
  
  and
  
      W[f_1,f_2]=2iξ,   W[f_1,f_3]=W[f_2,f_3]=0,and W[f_3,f̃_4]=-2μ.
  
  
  The construction of these solutions runs parallel to the constructions appearing in <cit.>, with the main differences arising from the presence of the jump condition atx=0.  For the sake of completeness, we have included the details in Appendix <ref>.  With Lemma <ref> in hand, we may proceed towards our main goal.
  
   We define
  
      f_4(x,ξ)=f̃_4(x,ξ)-c_1(ξ)f_1(x,ξ)-c_2 f_2(x,ξ)
  
  for ξ≠ 0, where 
  
      c_1(ξ)=-12iξW[f_2,f̃_4]andc_2(ξ)=12iξW[f_1,f̃_4].
   The solutions f_1,…,f_4 obey the following:
  
      W[f_1,f_2]=2iξ,   W[f_3,f_4]=-2μ,
  
  and 
  
      W[f_1,f_3]=W[f_1,f_4]=W[f_2,f_3]=W[f_2,f_4]=0.
   As the potential appearing in the operator ℋ is even, we may introduce
  
      g_j(x,ξ):=f_j(-x,ξ),   j∈{1,2,3,4},
  
  which are then solutions to (<ref>) with the same asymptotic behavior as x→-∞ as f_j when x→+∞. 
  
  
  We next introduce some matrix-valued solutions to (<ref>). 
  
   For j∈{1,2}, we define F_j=F_j(x,ξ) and G_j=G_j(x,ξ) via
  
      F_1 = [f_1 f_3],   F_2=[f_2  f_4],   G_1=[g_2 g_4],   G_2=[g_1 g_3].
  
  
  The solutionsF_jandG_jare connected, as the following lemma shows.  The proof is the same as the argument as in <cit.>.  
  
   For any ξ∈, there exist 2× 2 matrices A(ξ) and B(ξ) such that
  
      F_1(x,ξ)=G_1(x,ξ)A(ξ)+G_2(x,ξ)B(ξ).
  
  Furthermore, we have A(-ξ)=A(ξ), B(-ξ)=B(ξ), and
  
      G_2(x,ξ)=F_2(x,ξ)A(ξ)+F_1(x,ξ)B(ξ).
  
  
  
  
  
  
  
   The matrices A and B play a role similar to 
  the transmission and reflection coefficients of H (see (<ref>) below, for example).
  
  
  We would like to see that{f_1,f_3,g_1,g_3}is a basis for the set of solutions to (<ref>).  With this in mind, we define the following.
  
   We set
  
      D(ξ):=𝒲[F_1,G_2](ξ).
  
  
  The next result provides the essential connection between the spectral analysis and the scattering theory ofℋ. 
  
   For ξ≠ 0, the following are equivalent:
  
    (i)D(ξ)=0.
  
    (ii)12ξ^2+ω is an eigenvalue of $̋.
  
    (iii) A(ξ)=0.
  
  Forξ=0, the following are equivalent:
  
    (i')D(0)=0.
  
    (ii')ω is a resonance or eigenvalue of $̋. 
   As the proofs are the same as in <cit.> and <cit.>, we will only recall a few of the main components of the proof. 
  
  The equivalence of (i) and (iii) is a consequence of the 
  fact that
  
      D(ξ)=𝒲[F_1,G_2]=^t𝒲[F_1,G_2]=
      [[ 2iξ   0;   0 -2μ ]]A(ξ),
  
  which can be shown exactly as in <cit.>. 
  
  Let us prove (iii)(ii). Assume that A(ξ)=0. 
  Then, there exists 0≠ v(ξ)∈^2 
  such that A(ξ)v(ξ)=0. 
  By the same argument as that in  <cit.>, we have 
  for ξ≠0, 
  
      -2iξ P   =   -2iξ A(ξ)^∗PA(ξ)
      -2μ A(ξ)^∗QB(ξ)
            +2μ B(ξ)^∗QA(ξ)
      +2iξ B(ξ)^∗PB(ξ),
  
  where 
  
      P=[[ 1 0; 0 0 ]]and 
      Q=[[ 0 0; 0 1 ]].
   
  From this identity, we see -2iξ⟨ Pv(ξ),v(ξ)⟩
  =2iξ|PB(ξ)v(ξ)|^2. Since ξ≠0, we find that 
  Pv(ξ)=PB(ξ)v(ξ)=0. 
  On the other hand, by Lemma <ref>, we see 
  
      F_1(x,ξ)v(ξ)=G_1(x,ξ)A(ξ)v(ξ)+G_2(x,ξ)B(ξ)v=G_2(x,ξ)B(ξ)v(ξ).
  
  Since Pv(ξ)=0 and PB(ξ)v(ξ)=0, the left-hand side of (<ref>) belongs to L^2(0,∞) and the right-hand side of (<ref>) belongs to L^2(-∞,0).  Thus F_1(x,ξ)v(ξ) is a nontrivial L^2 solution of (<ref>), which yields (ii). 
  
  For the remaining implications, we again refer the reader to <cit.>. 
  
  In particular, in light of (<ref>) and the assumption thatωsatisfies the spectral condition in Definition <ref>, we have the following.
  
   For all ξ∈, D(ξ)≠ 0.  Moreover, for any ξ∈, 
  
      A(ξ)^-1 = D(ξ)^-1[[ 2iξ   0;   0 -2μ ]].
   
  
  We now introduce the following matrix-valued functions.
  
   We define
  
      (x,ξ)    = F_1(x,ξ)A(ξ)^-1[[ 1; 0 ]] = 2iξ F_1(x,ξ)D(ξ)^-1[[ 1; 0 ]], 
      (x,ξ)   =G_2(x,ξ)A(ξ)^-1[[ 1; 0 ]] = 2iξ G_2(x,ξ)D(ξ)^-1[[ 1; 0 ]].
  
  
  These functions will be essential in establishing the representation formula in Proposition <ref>.  We first show that bounded solutions to (<ref>) may be expressed in terms ofand.
  
   If f is a bounded solution to (<ref>), then there exist α(ξ) and β(ξ) such that
  
      f(x,ξ)=α(ξ)(x,ξ)+β(ξ)(x,ξ)
  
  for x≠ 0.
   By Corollary <ref> (which yields D(ξ)≠ 0), we have that {f_1,f_3,g_1,g_3} is a basis for the set of solutions to (<ref>).  Thus we may write f in the form
  
      f(x,ξ)=F_1(x,ξ)[[ d_1(ξ); d_2(ξ) ]] + G_2(x,ξ)[[ d_3(ξ); d_4(ξ) ]].
  
  Combining this with Lemma <ref> (i.e. expanding F_1 in terms of G_1 and G_2) and sending x→-∞, we can deduce (using the assumption that f is bounded) that
  
      A(ξ)[[ d_1(ξ); d_2(ξ) ]]=α(ξ)[[ 1; 0 ]]
  
  for some α(ξ).  Similarly (expanding G_2 in terms of F_1 and F_2 and sending x→∞) we can deduce
  
      A(ξ)[[ d_3(ξ); d_4(ξ) ]] = β(ξ)[[ 1; 0 ]]
  
  for some β(ξ).  This implies the result. 
  
  We will also need to understand the asymptotic behavior ofand. The starting point is to understand the asymptotic behavior of the matricesA(ξ)andB(ξ).
  
  
  For all k∈{1,2,…}, we have the following as |ξ|→∞: 
  
      A(ξ)=I+𝒪(|ξ|^-1)and∂_ξ^k A(ξ)=
      
      𝒪(|ξ|^-1),
         B(ξ) = 𝒪(|ξ|^-1)and∂_ξ^k B(ξ)=
      
      𝒪(|ξ|^-1),
  
  where the implicit constants depend continuously on ω. The key is to analyze D(ξ)=𝒲[F_1,G_2](ξ), which is connected to the matrix A(ξ) via (<ref>) above.  Similarly, analysis of the Wronskian 𝒲[F_1,G_1](ξ) provides the desired information about B(ξ).
  
  We begin by writing
  
      𝒲[F_1,G_2] = [[ W[f_1,g_1] W[f_1,g_3]; W[f_3,g_1] W[f_3,g_3] ]].
   
  We now utilize the estimates appearing in Lemma <ref> (at x=0) to derive that
  
      W[f_1,g_1]=2iξ+𝒪(1),  ∂_ξ W[f_1,g_1]=
      𝒪(1),  ∂_ξ^j W[f_1,g_1]= 
      𝒪(1)
  
  for j≥ 2.  Similarly we obtain the estimates
  
      W[f_1,g_3](ξ)=𝒪(1),  ∂_ξ^j 
      W[f_1,g_3](ξ)=
      𝒪(1)forj≥ 1,
         W[f_3,g_1](ξ)=𝒪(1),  ∂_ξ^j W[f_3,g_1](ξ)=
      𝒪(1)forj≥ 1, 
         W[f_3,g_3]=-2μ+𝒪(1),  ∂_ξ^jW[f_3,g_3] = ∂_ξ^j(-2μ)+𝒪(μ^-j)
      forj≥ 1,
  
  where we recall μ=√(ξ^2+4ω).
  
  Putting all of the pieces together, we have
  
      𝒲[F_1,G_2]=[[ 2iξ   0;   0 -2μ ]]  + 𝒪(1),
  
  and additionally
  
      ∂_ξ^j 𝒲[F_1,G_2] =
      𝒪(1)
  
  for j≥ 1. Combining this with (<ref>), we deduce that A(ξ)=I+𝒪(μ^-1).  The fact that we can differentiate the approximation above then allows us to deduce ∂_ξ^k A(ξ)=
  𝒪(|ξ|^-1), as desired. 
  
  A similar argument based on the fact that
  
      𝒲[F_1,G_1](ξ)=[[ W[f_1,g_2] W[f_1,g_4]; W[f_3,g_2] W[f_3,g_4] ]]=-^t B(ξ)[[ 2iξ   0;   0 -2μ ]]
  
  yields to the desired estimates for B.  For the second equality above, we refer the reader to <cit.>. 
  
  We turn to the behavior ofand. We first introduce two scalar functionsT̃(ξ)andR̃(ξ)(the transmission and reflection coefficients).
  
   We define T̃(ξ) and R̃(ξ) by imposing
  
      2iξ[[ 1 0; 0 0 ]]D(ξ)^-1[[ 1; 0 ]]    =:[[ T̃(ξ);     0 ]], 
      
       2iξ[[ 1 0; 0 0 ]]B(ξ)D(ξ)^-1[[ 1; 0 ]]     =: [[ R̃(ξ);     0 ]].
    Using (<ref>) and Lemma <ref>, we may deduce that T̃ and R̃, along with their derivatives, are bounded functions. For ℓ∈{0,1,2,…}, we have
  
      ∂_ξ^ℓ((x,ξ) -T̃(ξ)e^ixξ[[ 1; 0 ]])   ≲μ^-1 x^ℓ+1e^-min{γ,μ}x)asx→∞,
      ∂_ξ^ℓ((x,ξ)-(e^ixξ+R̃(ξ)e^-ixξ)[[ 1; 0 ]])   ≲μ^-1|x|^ℓ+1e^min{γ,μ}xasx→-∞, 
      ∂_ξ^ℓ((x,ξ)-T̃(ξ)e^-ixξ[[ 1; 0 ]])    ≲μ^-1|x|^ℓ+1e^min{γ,μ}xasx→-∞,
      ∂_ξ^ℓ((x,ξ)-(e^-ixξ+R̃(ξ)e^ixξ)[[ 1; 0 ]])   ≲μ^-1x^ℓ+1e^-min{γ,μ}xasx→∞,
  
  with implicit constants depending continuously on ω.
   We focus on describing  in the region x≥ 0.  Similar arguments treat the remaining cases.
  
  By definition, we may write
  
      = 2iξ[f_1 0]D(ξ)^-1[[ 1; 0 ]] + 2iξ[0 f_3]D(ξ)^-1[[ 1; 0 ]].
  
  We now insert the approximations to f_1 and f_3 from Lemma <ref>.  The contribution of f_1 is then
  
      T̃(ξ)e^ixξ[[ 1; 0 ]] + 2iξ e^ixξ[ J_1 [ 0; 0 ]]D(ξ)^-1[[ 1; 0 ]],
  
  where 
  
      J_1(x,ξ)=e^-ixξf_1(x,ξ)-[[ 1; 0 ]].
  
  Similarly, the contribution of f_3 is 
  
      2iξ e^-μ x[[ 0 0; 0 1 ]]D(ξ)^-1[[ 1; 0 ]] + 2iξ e^-μ x[[ 0; 0 ] J_3] D(ξ)^-1[[ 1; 0 ]],
  
  where 
  
      J_3(x,ξ)=e^μ xf_3(x,ξ)-[[ 0; 1 ]].
    
  
  Our task is to estimate the second term in (<ref>) and both terms in (<ref>).  The key ingredients are the estimates in Lemma <ref>, as well as Lemma <ref> (where we recall the connection between A(ξ) and D(ξ) given by Corollary <ref>).  As the estimates are all similar, let us only present just the estimate of the second term in (<ref>). 
  
  Given ℓ≥ 0, we estimate 
  
      ∑_i+j+k=ℓ   | ∂_ξ^i e^ixξ| | [∂_ξ^j J_1 [ 0; 0 ]]|  | ∂_ξ^k[2iξ D(ξ)^-1][[ 1; 0 ]]| 
      
         ≲∑_i+j+k=ℓ |x|^i μ^-1⟨ x⟩^j+1e^-min{γ,μ}x
      
      ≲μ^-1⟨ x⟩^ℓ+1 e^-min{γ,μ}x,
  
  as desired. 
  
  
  With the particular solutions constructed above, we can construct the integral kernel of the resolvent:
   For ξ≠ 0, 
  
      [-̋12ξ^2-ω-i0]^-1(x,y)     =  -2F_1(x,ξ)D^-1(ξ)^tG_2(y,ξ)σ_3     x≥ y, 
       -2G_2(x,ξ)D^-1(ξ)^tF_1(y,ξ)σ_3     x≤ y, 
      
      [-̋12ξ^2-ω+i0]^-1(x,y)     =  -2F_1(x,-ξ)D^-1(-ξ)^tG_2(y,-ξ)σ_3     x≥ y, 
       -2G_2(x,-ξ)D^-1(-ξ)^t F_1(y,-ξ)σ_3     x ≤ y.
   It suffices to prove the first identity.  We introduce h=[-̋12ξ^2-ω-i0]^-1φ, i.e. the solution to [-̋12ξ^2-ω-i0]h=φ.  By variation of parameters, we may write
  
      h(x,ξ)=F_1(x,ξ)[[ d_1(x,ξ); d_2(x,ξ) ]] + G_2(x,ξ)[[ d_3(x,ξ); d_4(x,ξ) ]],
  
  where (denoting ddx by ')
  
      F_1(x,ξ)[[ d_1'(x,ξ); d_2'(x,ξ) ]] + G_2(x,ξ)[[ d_3'(x,ξ); d_4'(x,ξ) ]] = [[ 0; 0 ]]
  
  and
  
      [[  F_1(x,ξ)  G_2(x,ξ); F_1'(x,ξ) G_2'(x,ξ) ]][[ d_1'(x,ξ); d_2'(x,ξ); d_3'(x,ξ); d_4'(x,ξ) ]] = [[     0;     0; -2φ_1;  2φ_2 ]]=[[      0;      0; -2σ_3φ ]]
  
  As D(ξ)≠ 0, we can invert the matrix appearing on the left (cf. <cit.>), which leads to
  
      [[ d_1'(x,ξ); d_2'(x,ξ); d_3'(x,ξ); d_4'(x,ξ) ]]     = [[    -^t D^-1(ξ)G_2'(x,ξ)   ^t D^-1(ξ)^t G_2(x,ξ); ^t D^-1(ξ) ^t F_1'(x,ξ)  -^t D^-1(ξ)^t F_1(x,ξ) ]][[      0;      0; -2σ_3φ ]] 
          = [[ -2^t D^-1(ξ) ^tG_2(x,ξ)σ_3 φ;   2^tD^-1(ξ)^t F_1(x,ξ)σ_3 φ ]].
  
  In particular, integrating in x yields
  
      [[ d_1(x,ξ); d_2(x,ξ) ]]   =-2^tD^-1(ξ)∫_-∞^x ^tG_2(y,ξ)σ_3φ(y) dy, 
      [[ d_3(x,ξ); d_4(x,ξ) ]]   =-2^tD^-1(ξ)∫_x^∞^tF_1(y,ξ)σ_3φ(y) dy.
   
  This implies the desired identity. 
  
  With the integral kernel for the resolvent in place, we can now compute the spectral measure.
  
   We define E=E(x,ξ) by
  
      E = [ ],
  
  where  and  are as in Definition <ref>.
   For ξ∈,
  
      [-̋12ξ^2-ω-i0]^-1(x,y)-[-̋12ξ^2-ω+i0]^-1(x,y)=-1iξE(x,ξ)E(y,ξ)^*σ_3,
  
  where σ_3 is as in (<ref>). 
   We proceed as in <cit.>. We set
  
      K(x,y,ξ)=LHS(<ref>).
  
  Now, for fixed y, K(x,y,ξ) is a bounded solution to (<ref>), so that by Lemma <ref> we may write
  
      K(x,y,ξ)=E(x,ξ)M(y,ξ)for some M∈^2× 2.
  
  Using Lemma <ref>, we also see that
  
      σ_3K(x,y,ξ)^*σ_3 = -K(y,x,ξ),
  
  which (after some rearranging) implies
  
      C(ξ):=E(y,ξ)^-1σ_3 M(y,ξ)^*=-M(x,ξ)σ_3[E(x,ξ)^*]^-1.
  
  Observing that C(ξ)^*=-C(ξ)^-1, we see that C(ξ) takes the form
  
      C(ξ)=[[  iα   β; -β̅   i ]]
  
  for some α,∈ and β∈. We therefore obtain
  
      K(x,y,ξ)=-E(x,ξ)C(ξ)E(y,ξ)^*σ_3.
  
  
  To complete the proof we must show C(ξ)=-1iξI, i.e. α==-1ξ and β=0.  To this end, we will use two different expressions for K(x,y,ξ) and send x→∞ and y→-∞.  First, using Lemma <ref> and (<ref>) in the original definition of K, we deduce that 
  
      K(x,y,ξ)     = [1iξT̃(ξ)e^i(x-y)ξ+1iξT̃(-ξ)e^-i(x-y)ξ][[ 1 0; 0 0 ]] +o(1)
  
  as x→∞ and y→-∞.  On the other hand, using (<ref>) and the asymptotic behavior given in Lemma <ref>, we deduce
  
      K(x,y,ξ)     = -[{β |T̃(ξ)|^2-β̅|R̃(ξ)|^2+iαT̃(ξ)R̃(-ξ)+iT̃(-ξ)T̃(ξ)} e^i(x+y)ξ
              -β̅e^-i(x+y)ξ + {iαT̃(ξ)-β̅R̃(ξ)}e^i(x-y)ξ
              +{iT̃(-ξ)-β̅R̃(-ξ)} e^-i(x-y)ξ][[ 1 0; 0 0 ]] + o(1)
  
  as x→∞ and y→-∞.  It follows that β=0, while α==-1ξ, as desired.  
  
  
  Finally, we can obtain the following representation formula.
  
   We define _±(x,ξ) and _±(x,ξ) via
  
      _+=,      F_-=σ_1, 
      _+=,     _-=σ_1,
  
  where σ_1 is as in (<ref>), and we set
  
      e_±(x,ξ) = _±(x,ξ)    ξ≥ 0, 
      _±(x,-ξ)    ξ≤ 0.
   With e_±=e_±(x,ξ) as above, we have 
  
      ⟨ e^itP_c φ,ψ⟩    =12πe^itω∫ e^itξ^2/2⟨φ,σ_3 e_+(·,ξ)⟩⟨ψ,e_+(·,ξ)⟩ dξ
            + 12π e^-itω∫ e^-itξ^2/2⟨φ,σ_3 e_-(·,ξ)⟩⟨ψ,e_-(·,ξ)⟩ dξ.
  
  
  
  
  
  
   Using Lemma <ref>, we write
  
      [-̋   12ξ^2-ω-i0]^-1(x,y)-[-̋12ξ^2-ω+i0]^-1(x,y) 
          = -1iξ[e_+(x,ξ) e_+(x,-ξ)][e_+(y,ξ) e_+(y,-ξ)]^*σ_3.
  
  We also claim that
  
      [+̋12ξ^2+ω-i0]^-1(x,y)-[+̋12ξ^2+ω+i0]^-1(x,y) 
            = -1iξ[e_-(x,ξ) e_-(x,-ξ)][e_-(y,ξ) e_-(y,-ξ)]^*σ_3.
  
  Inserting these relations into the expression above and simplifying then leads to the desired formula (<ref>).  To derive (<ref>), we again use Lemma <ref>, along with the identity
  
      +̋12ξ^2 + ω = -σ_1[H-12ξ^2 - ω]σ_1.
  
  Inserting these identities for the integral kernel into (<ref>) and (<ref>) now leads to the representation formula.  The L^2→ L^2 estimate (<ref>) now follows from Proposition <ref> and Lemma <ref>. In the next sections, we will prove (<ref>) and (<ref>) by considering separately the low-energy and high-energy contributions.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Estimates for low energies
  
  
  In this section, we estimate the contribution of the low energy regime to the dispersive estimates (<ref>) and (<ref>).   To implement the cutoff to low and high frequencies, we letχbe a smooth, even bump function such thatχ(ξ)=1for|ξ|≤λ, andχ(ξ)=0for|ξ|≥2λ.  We will chooseλ=C𝒱_L^1for some sufficiently largeC. 
  
  Recalling the representation formula (<ref>) and Lemma <ref>, we have the following: LetP_c^+andP_c^-be projections to positive part and negative part of the absolutely continuous 
  spectrum ofℋ. Then 
  
      2π i⟨ e^itℋχ(ℋ-ω)
      P_c^+φ,ψ⟩
         =   
      e^itω∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      ⟨{(ℋ-ξ^22-ω-i0)^-1
      -(ℋ-ξ^22-ω+i0)^-1}φ,ψ⟩ 
      dξ
         =   
      -2e^itω∫∫_x≥ y∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      F_1(x,ξ)D^-1(ξ)^tG_2(y,ξ)σ_3φ(y)ψ(x)
       dξ  dx dy
            
      +2e^itω∫∫_x≤ y∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      G_2(x,ξ)D^-1(ξ)^tF_1(y,ξ)σ_3φ(y)ψ(x)
       dξ dx dy,
  
  with a similar formula fore^itℋχ(ℋ-ω)P_c^-. 
  
  Because the functionsF_j,G_j, andD^-1have the same decay/regularity properties as the analogous functions appearing in <cit.>, we can follow much of the presentation in <cit.>.  Accordingly, we shall keep our presentation rather brief. 
  
  It is sufficient to consider only first term on the right-hand side of (<ref>).  We split the region of the integral into the three subregions:x≥ y≥ 0,x≥ 0≥ yand0≥ x≥ y.  By symmetry, it suffices to consider the first two regions.
  
  For the casex≥ 0≥ y, we split theξintegral as follows: 
  
      ∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      F_1(x,ξ)D^-1(ξ)^tG_2(y,ξ)σ_3dξ
         =   ∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      [f_1(x,ξ)  [ 0; 0 ]]
      D^-1(ξ)
      [[ ^tf_1(-y,ξ);       0   0 ]]
      σ_3dξ
            
      +∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      [f_1(x,ξ)  [ 0; 0 ]]
      D^-1(ξ)
      [[       0   0; ^tf_3(-y,ξ) ]]
      σ_3dξ
            +∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      [ [ 0; 0 ]f_3(x,ξ)]
      D^-1(ξ)
      [[ ^tf_1(-y,ξ);       0   0 ]]
      σ_3dξ
            +∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      [[ 0; 0 ]f_3(x,ξ)]
      D^-1(ξ)
      [[       0   0; ^tf_3(-y,ξ) ]]
      σ_3dξ
         =:   I_1(x,y)+I_2(x,y)+I_3(x,y)+I_4(x,y).
  
  Let us show how to estimateI_1; the remaining terms are similar.  Writingχ̃for a cut-off function such thatχ̃χ=χ, we have
  
      |I_1(x,y)|
         ≲   t^-1/2∫_-∞^∞
      |[ξχ(ξ^2)D^-1(ξ)]∗
      [χ̃(ξ^2)e^-ixξf_1(x,ξ)]∗[χ̃(ξ^2)e^iyξf_1(-y,ξ)]
      (η)|dη
         ≲   t^-1/2[ξχ(ξ^2)D^-1(ξ)]_L^1[χ̃(ξ^2)e^-ixξf_1(x,ξ)]_L^1[χ̃(ξ^2)e^iyξf_1(-y,ξ)]_L^1.
  
  
  Using Lemmas <ref> and <ref>, we deduce that
  
      [ξχ(ξ^2)D^-1(ξ)]_L^1≲ 1
  
  and
  
      sup_x≥ 0[χ̃(ξ^2)e^-ixξf_1(x,ξ)]_L^1≲sup_x≥ 0(1-_ξ^2)
      {χ̃(ξ^2)e^-ixξf_1(x,ξ)}_L^2≲ 1.
  
  Similarly, 
  
      sup_y≤0[χ̃(ξ^2)e^iyξf_1(-y,ξ)]_L^1≲ 1.
  
  Hence we have 
  
      sup_x≥ 0, y≤0|I_1(x,y)|≲ t^-1/2.
  
  
  For the case0≥ x≥ y, using Lemma <ref>, we find
  
      ∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      F_1(x,ξ)D^-1(ξ)^tG_2(y,ξ)σ_3dξ
         =   ∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      G_1(x,ξ)A(ξ)D^-1(ξ)^tG_2(y,ξ)σ_3dξ
            +∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      G_2(x,ξ)B(ξ)D^-1(ξ)^tG_2(y,ξ)σ_3dξ
         =:   J(x,y)+K(x,y).
  
  Again, we splitJinto the following four pieces.
  
      J(x,y)   =   ∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      [f_2(-x,ξ)  [ 0; 0 ]]
      A(ξ)D^-1(ξ)
      [[ f_1(-y,ξ);     0   0 ]]
      σ_3dξ
            
      +∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      [f_2(-x,ξ)  [ 0; 0 ]]
      A(ξ)D^-1(ξ)
      [[     0   0; f_3(-y,ξ) ]]
      σ_3dξ
            +∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      [ [ 0; 0 ]f_4(-x,ξ)]
      A(ξ)D^-1(ξ)
      [[ f_1(-y,ξ);     0   0 ]]
      σ_3dξ
            +∫_0^∞
      e^i/2tξ^2ξχ(ξ^2)
      [[ 0; 0 ]f_4(-x,ξ)]
      A(ξ)D^-1(ξ)
      [[     0   0; f_3(-y,ξ) ]]
      σ_3dξ
         =:   J_1(x,y)+J_2(x,y)+J_3(x,y)+J_4(x,y).
  
  Using Corollary <ref>, we may deduce thatJ_2≡ J_3≡ 0.   Arguing as in the casex≥ 0≥ y, we can then obtain
  
      sup_0≥ x≥ y|J(x,y)|≲sup_0≥ x≥ y(|J_1(x,y)|+|J_4(x,y)|)≲ t^-1/2,
  
  with a similar estimate forK. 
  
  Collecting the above estimates, we conclude 
  
      |2π i⟨ e^itℋχ(ℋ-ω)
      P_cφ,ψ⟩|   ≲   
      t^-1/2φ_L^1ψ_L^1,
  
  with implicit constant depending on𝒱.  This completes the estimation of the contribution of the low energies to the unweighted dispersive estimate (<ref>). 
  
  For the low-energy contribution to the weighted estimate (<ref>), we will once again proceed essentially as in <cit.> (see <cit.>) due to the fact that the distorted plane wavese_±constructed above share the same decay/regularity properties as those appearing in <cit.>.
  
  By Proposition <ref>, we have
  
      2π i
      ⟨ e^itℋχ(ℋ-ω)
      P_c^+φ,ψ⟩
         =   
      -4ie^itω∫_0^∞
      e^i/2tξ^2ξ^2χ(ξ)
      ⟨ f,σ_3F_1(·,ξ)D(ξ)^-1e⟩
      
                                    ×⟨ g,σ_3F_1(·,ξ)D(ξ)^-1e⟩  dξ
            
      -4ie^itω∫^0_-∞
      e^i/2tξ^2ξ^2χ(ξ)
      ⟨ f,σ_3G_2(·,-ξ)D(-ξ)^-1e⟩
      
                                    ×⟨ g,σ_3G_2(·,-ξ)D(ξ)^-1e⟩  dξ.
  
  Integrating by parts via the identity
  
      e^i/2tξ^2=_ξ(e^i/2tξ^2)/itξ,
  
  we have
  
      2π i
      ⟨ e^itℋ(1-χ(ℋ-ω))
      P_c^+φ,ψ⟩
         =   4/te^itω∫_0^∞
      e^i/2tξ^2_ξ{ξχ(ξ)
      ⟨ f,σ_3F_1(·,ξ)D(ξ)^-1e⟩
                                    ×⟨ g,σ_3F_1(·,ξ)D(ξ)^-1e⟩}dξ
            
      +4/te^itω∫^0_-∞
      e^i/2tξ^2_ξ{ξχ(ξ)
      ⟨ f,σ_3G_2(·,-ξ)D(-ξ)^-1e⟩
                                    ×⟨ g,σ_3G_2(·,-ξ)D(-ξ)^-1e⟩} dξ
         =:   L_1+L_2.
  
  
  We focus our attention onL_1, asL_2can be treated in a similar way.  We splitL_1into the following two pieces: 
  
      L_1
         =   4/te^itω∫_0^∞
      e^i/2tξ^2_ξ(ξχ(ξ))
      ⟨ f,σ_3F_1(·,ξ)D(ξ)^-1e⟩
                                    ×⟨ g,σ_3F_1(·,ξ)D(ξ)^-1e⟩ dξ
            +4/te^itω∫_0^∞
      e^i/2tξ^2ξχ(ξ)
      _ξ{⟨ f,σ_3F_1(·,ξ)D(ξ)^-1e⟩
                                    ×⟨ g,σ_3F_1(·,ξ)D(ξ)^-1e⟩}dξ
         =:   L_1,1+L_1,2.
  
  
  Let us now focus now on the termL_1,1; we refer the reader to <cit.> for details concerning the remaining terms.  We letω(ξ)=_ξ(ξχ(ξ))and 
  letω̃(ξ)be an another smooth 
  cut-off function satisfyingω=
  ωω̃. 
  By theL^1-L^∞dispersive estimate for the free Schrödinger group, we see
  
      |L_1,1|
      
      
      
      
      
      
      
         ≲   t^-3/2ℱ̃_ξ↦ u^-1
      [ω(ξ)⟨ f,σ_3F_1(·,ξ)D(ξ)^-1e⟩ ]_L_u^1
                    ×ℱ̃_ξ↦ u^-1
      [ω̃(ξ)⟨ g,σ_3F_1(·,ξ)D(ξ)^-1e⟩ ]_L_u^1,
  
  whereℱ̃is the Fourier 
  cosine transform. If we can prove
  
      ℱ̃_ξ↦ u^-1
      [ω(ξ)F_1(·,ξ)D(ξ)^-1e]_L_u^1≲⟨ x⟩,
  
  then we can obtain 
  
      ℱ̃_ξ↦ u^-1
      [ω(ξ)⟨ f,σ_3F_1(·,ξ)D(ξ)^-1e⟩ ]_L_u^1≲⟨ x⟩ f_L_x^1.
  
  This bound, along with with a similar estimate for 
  
      ℱ̃_ξ↦ u^-1
      [ω̃(ξ)⟨ g,σ_3F_1(·,ξ)D(ξ)^-1e⟩ ],
  
  will yield the desired decay estimate (<ref>). 
  
  We turn to (<ref>).  For the casex≥0, an integration by 
  parts implies 
  
      ℱ̃_ξ↦ u^-1
      [ω(ξ)F_1(·,ξ)D(ξ)^-1e](u)
         =   ∑_±{
      -1/2i(x± u)ω(0)f_1(x,0)D(0)^-1_11
            -1/2i(x± u)∫_0^∞e^i(x± u)ξ_ξ{ω(ξ)e^-ixξf_1(x,ξ)D(ξ)^-1_11}dξ}
            -1/u∫_0^∞sin(ξ u)
      _ξ{ω(ξ)f_3(x,ξ)D(ξ)^-1_21}dξ,
  
  whereD(ξ)^-1_ijis the(i,j)component ofD(ξ)^-1. Integrating by parts in the above identity and using Lemma <ref>, we have
  
      |ℱ̃_ξ↦ u^-1
      [ω(ξ)F_1(·,ξ)D(ξ)^-1e](u)|
      ≲|x|/|x^2-u^2|+1/(u+x)^2+1/(u-x)^2+1/u^2.
  
  Combining this with the estimate
  
      |ℱ̃_ξ↦ u^-1
      [ω(ξ)F_1(·,ξ)D(ξ)^-1e](u)|
      ≲1,
  
  we obtain (<ref>). 
  
  For the casex≤0the key is to rewriteF_1(x,ξ)D^-1(ξ)ein terms off_1(-x,ξ)andf_3(-x,ξ): 
  
      F_1(x,ξ)D^-1(ξ)e
         =   f_1(-x,-ξ)-f_1(-x,ξ)/2iξ
      +R̃(ξ)+1/2iξf_1(-x,ξ)
            +f_3(-x,ξ)(B(ξ)D^-1(ξ))_21.
  
  Now observe that the second term in the right hand side of (<ref>) is smooth for anyξ∈, due to the fact thatR̃(0)=-1(see Lemma <ref>). Substituting (<ref>) intoℱ̃_ξ↦ u^-1 [ω(ξ)F_1(·,ξ)D(ξ)^-1e](u)and applying an  integration by parts and Lemma <ref> to the resulting expression, we obtain (<ref>).  We again refer the reader to <cit.> for more complete details. 
  
  Collecting the above estimates, we arrive at the estimate
  
      ⟨ e^itℋ(1-χ(ℋ-ω))
      P_cφ,ψ⟩≲ t^-3/2⟨ x⟩ f_L^1⟨ x⟩ g_L^1,
  
  with some implicit constant depending on𝒱, which completes the estimation of the contribution of the low energies  to the weighted dispersive estimate (<ref>). 
  
  
  
  
   §.§ Estimates for high energies
  
  
  In this section, we obtain estimates for the high-energy contributions by using the Born series expansion 
  
      (ℋ-ξ^22-ω∓ i0)^-1
         =   ∑_n=0^∞(-1)^n
      (ℋ_0-ξ^22-ω∓ i0)^-1{𝒱(ℋ_0-ξ^22-ω∓ i0)^-1}^n.
  
  In particular, we write 
  
      2π i⟨ e^itℋ(1-χ(ℋ-ω))
      P_cφ,ψ⟩
         =   
      e^itω∑_n=0^∞(-1)^n∫_0^∞
      e^i/2tξ^2ξ[1-χ(ξ^2)]
                ×[
      ⟨{
      (ℋ_0-ξ^22-ω- i0)^-1{𝒱(ℋ_0-ξ^22-ω- i0)^-1}^n
      }φ,ψ⟩.
                  -.
      ⟨{
      (ℋ_0-ξ^22-ω+ i0)^-1{𝒱(ℋ_0-ξ^22-ω+ i0)^-1}^n
      }φ,ψ⟩] dξ
         =:   e^itω∑_n=0^∞(-1)^nI_n.
  
  
  To obtain decay estimates forI_n, we use that 
  the integral kernel of(ℋ_0-ξ^2/2-ω∓ i0)^-1is given by 
  
      [
      [  (H-ξ^2/2∓ i0)^-1(x,y)                      0;                      0 -(H+μ^2/2± i0)^-1(x,y) ]],
  
  whereμ=√(ξ^2+4ω)and 
  
      (H-ξ^22∓ i0)^-1(x,y)
         =   ∓iξ{e^∓ iξ|x-y|
      +R(∓ξ)e^∓ iξ(|x|+|y|)}
         =:   K^1_∓(x,y,ξ),
      
      (H+μ^22± i0)^-1(x,y)
         =   -1μ{e^-μ|x-y|
      +R(iμ)
      e^-μ(|x|+|y|)}
         =:   K^2(x,y,ξ).
  
  HereR(ξ)=qiξ-qis the reflection coefficient associated to the operatorH=-12∂_x^2+qδ_0. 
  
  We first evaluateI_0. A direct computation shows  
  
      I_0   =   ∫_0^∞
      e^i/2tξ^2ξ[1-χ(ξ^2)]
              ×[
      ∫_-∞^+∞{∫_-∞^+∞[K^1_-(x,y,ξ)-K^1_+(x,y,ξ)]φ_1(y)dy}ψ_1(x)dx
      ]dξ
         =   ∫_-∞^+∞∫_-∞^+∞{∫_0^∞
      e^i/2tξ^2ξ[1-χ(ξ^2)]
      [K^1_-(x,y,ξ)-K^1_+(x,y,ξ)]dξ}
                                                    ×ψ_1(x)φ_1(y) dx dy,
  
  whereφ=[[  φ_1; φ̅_1 ]]andψ=[[  ψ_1; ψ̅_1 ]]. Hence if we can show that
  
      sup_x,y∈|∫_0^∞e^i/2tξ^2ξ[1-χ(ξ^2)]
      [K^1_-(x,y,ξ)-K^1_+(x,y,ξ)]dξ|≲ t^-1/2,
  
  then we will have the desired estimate 
  
      |I_0|≲ t^-1/2φ_L^1ψ_L^1.
  
  To obtain (<ref>), it suffices to show 
  
      sup_a∈|∫_0^∞e^iaξ+i/2tξ^2[1-χ(ξ^2)]g(ξ)
      dξ|≲ t^-1/2,
  
  whereg=1, RorR̅. Using the identity 
  
      e^iaξ+i/2tξ^2
      =_ξ{(ξ+a/t)e^iaξ+i/2tξ^2}/1+it(ξ+a/t)^2
  
  and integrating by parts, we obtain
  
      ∫_0^∞e^iaξ+i/2tξ^2[1-χ(ξ^2)]g(ξ)
      dξ
         =   
      -∫_0^∞(ξ+a/t)e^iaξ+i/2tξ^2_ξ{[1-χ(ξ^2)]g(ξ)/1+it(ξ+a/t)^2}
      dξ
         =   
      -∫_0^∞(ξ+a/t)e^iaξ+i/2tξ^2_ξ{[1-χ(ξ^2)]g(ξ)}/1+it(ξ+a/t)^2dξ
            
      +2it∫_0^∞(ξ+a/t)^2e^iaξ+i/2tξ^2[1-χ(ξ^2)]g(ξ)/{1+it(ξ+a/t)^2}^2
      dξ.
  
  Thus we can estimate
  
      |∫_0^∞e^iaξ+i/2tξ^2[1-χ(ξ^2)]g(ξ)
      dξ|
         ≲   
      t^-1/2∫_0^∞|_ξ{[1-χ(ξ^2)]g(ξ)}|dξ
      +∫_0^∞dξ/1+t(ξ+a/t)^2
         ≲   t^-1/2,
  
  with implicit constants independent ofa.  This yields (<ref>), as desired.
  
  We turn to the estimation ofI_nforn≥1. For this, we rely on the following identity:
  
      (ℋ_q-ξ^22-ω∓ i0)^-1{𝒱(ℋ_q-ξ^22-ω∓ i0)^-1}^n
      }φ(x)
         =   ∫_^n+1𝒦_∓^n(x,x_1,⋯, x_n+1)φ(x_n+1)
      dx_1⋯ dx_n+1,
  
  where 
  
      𝒦_∓^n(x,x_1,⋯, x_n+1,ξ)
         =   [
      [ K_∓^1(x,x_1,ξ)              0;              0   K^2(x,x_1,ξ) ]]
            ×∏_j=1^n[
      [  V_1(x_j)  V_2(x_j); -V_2(x_j) -V_1(x_j) ]]
      [
      [ K_∓^1(x_j,x_j+1,ξ)                  0;                  0   K^2(x_j,x_j+1,ξ) ]],
  
  whereK^1_±andK^2are as above and we have setV_1(x)=σ(p/2+1)Q_ω^p(x)andV_2(x)=σp/2Q_ω^p(x). 
  
  It follows that
  
      |I_n|    ≲∑_±| ∫_0^∞∫_^n+2 e^i/2tξ^2ξ[1-χ(ξ^2)] 
             × K_∓^n(x,x_1,…,x_n+1,ξ)φ(x_n+1)ψ̅(x) dx dx_1⋯ dx_n+1 dξ|.
  
  We now observe that each component of the matrices𝒦_∓^n(x,x_1,⋯, x_n+1)is sum of2^nterms of the form
  
      C(p)∏_j=1^nV_1(x_j)ξ^-n-1
      g(ξ)e^ih(x,x_1,⋯,x_n+1)ξe^-k(x,x_1,⋯,x_n+1)μ,
  
  wheregis some power ofRorR̅, andhandkare real-valued functions withk(x,x_1,⋯,x_n+1)≥ 0.  Thus the right-hand side of the above inequality  
  is bounded by sum of2^nterms of the form
  
      ∫_^n+2
      |ψ(x)|∏_j=1^nV_1(x_j)|φ(x_n+1)| |
      I(x,x_1,⋯,x_n+1,ξ)|dx dx_1⋯ dx_n+1,
  
  where
  
      I(x,x_1,⋯,x_n+1,ξ)
         =   ∫_0^∞e^i/2tξ^2ξ^-n[1-χ(ξ^2)]
      g(ξ)e^ih(x,x_1,⋯,x_n+1)ξe^-k(x,x_1,⋯,x_n+1)μdξ.
  
  Thus, if we can prove that
  
      sup_a∈,b≥ 0|∫_0^∞e^i/2tξ^2e^iaξe^-bμξ^-n[1-χ(ξ^2)]
      g(ξ)dξ|≤ Cλ^-n+1t^-1/2,
  
  where the constantCis independent ofnandg, 
  then we can obtain the estimate 
  
      |I_n|≲ t^-1/22^n λ^-n+1𝒱_L^1^n
      φ_L^1ψ_L^1,
  
  with implicit constant independent ofn. 
  
  We turn to (<ref>). Using (<ref>) once again and integrating by parts, we obtain
  
      ∫_0^∞e^iaξ+i/2tξ^2
      e^-bμξ^-n[1-χ(ξ^2)]
      g(ξ)
      dξ
         =   
      -∫_0^∞(ξ+a/t)e^iaξ+i/2tξ^2_ξ{e^-bμξ^-n[1-χ(ξ^2)]g(ξ)/
      1+it(ξ+a/t)^2}dξ
         =   
      -∫_0^∞(ξ+a/t)e^iaξ+i/2tξ^2_ξ{e^-bμξ^-n[1-χ(ξ^2)]
      g(ξ)}/1+it(ξ+a/t)^2dξ
            
      +2it∫_0^∞(ξ+a/t)^2e^iaξ+i/2tξ^2e^-bμξ^-n[1-χ(ξ^2)]g(ξ)/{
      1+it(ξ+a/t)^2}^2dξ.
  
  Thus we firstly obtain 
  
      |∫_0^∞e^iaξ+i/2tξ^2
      e^-bμξ^-n[1-χ(ξ^2)]g(ξ)
      dξ|
         ≲   
      t^-1/2∫_0^∞|_ξ{e^-bμξ^-n[1-χ(ξ^2)]g(ξ)}|dξ
      +λ^-n∫_0^∞dξ/1+t(ξ+a/t)^2.
  
  
  To continue, we estimate as follows:
  
      ∫_0^∞|_ξ{e^-bμξ^-n[1-χ(ξ^2)]g(ξ)}|dξ
         ≤   
      b∫_0^∞e^-bμξ^-n[1-χ(ξ^2)]|g(ξ)|dξ
      +
      ∫_0^∞e^-bμ|_ξ{ξ^-n[1-χ(ξ^2)]g(ξ)}|dξ
         ≤   λ^-nb∫_0^∞e^-bμdξ
      +∫_0^∞|_ξ{ξ^-n[1-χ(ξ^2)]g(ξ)}|dξ
         ≲   λ^-n+1,
  
  where we note that the implicit constants are independent ofb.  In particular, continuing from above we obtain (<ref>). 
  
  Substituting (<ref>) and (<ref>) into (<ref>), we see that withλ=C𝒱_L^1, we obtain 
  
      |2π i⟨ e^itℋ(1-χ(ℋ-ω))
      P_cφ,ψ⟩|
         ≲   
      t^-1/2φ_L^1ψ_L^1∑_n=0^∞2^n λ^-n+1𝒱_L^1^n
      
         ≲   
      t^-1/2φ_L^1ψ_L^1,
  
  yielding the desired unweighted estimate (<ref>) in the high-energy regime.  
  
  For the high-energy contribution to the weighted estimate (<ref>), we begin by performing an integration by parts in (<ref>), obtaining
  
      2π i ⟨ e^itℋ(1-χ(ℋ-ω))
      P_cφ,ψ⟩
         =   
      t^-1e^itω∑_n=0^∞(-1)^n∫_0^∞
      e^i/2tξ^2_ξ[{[1-χ(ξ^2)]
                ×[
      ⟨{
      (ℋ_q-ξ^22-ω- i0)^-1{𝒱(ℋ_q-ξ^22-ω- i0)^-1}^n
      }φ,ψ⟩.
                  -.
      ⟨{
      (ℋ_q-ξ^22-ω+ i0)^-1{𝒱(ℋ_q-ξ^22-ω+ i0)^-1}^n
      }φ,ψ⟩] ]dξ.
  
  One may now proceed essentially as we did for the unweighted estimate, leading to the estimate 
  
      ⟨ e^itℋ(1-χ(ℋ-ω))
      P_cφ,ψ⟩≲ t^-3/2⟨ x⟩φ_L^1⟨ x⟩ψ_L^1.
  
  In particular, the additional decay is already present in the formula above, while the weights appear due to the fact that the additional derivative hits the resolvent, leading to terms like|x_j-x_j-1|(cf. the definition ofK_±^1andK^2above).  We then apply the simple bound|x_j-x_j-1|≤ |x_j|+|x_j-1|and use eitherV_1,φ, orψto absorb the weight. 
  
  This completes the estimation of the high-energy regime, and hence completes the proof of Proposition <ref>. 
  
  
  
  
   §.§ Interpolation estimates
  
  
  In Section <ref>, we will rely on the dispersive estimates appearing in Proposition <ref> in an essential way.  However, we will not apply these estimates directly, but rather in an interpolated form.  In this section, we collect the estimates in the form that we will need them.
  
  First, interpolating (<ref>) with (<ref>) leads to the estimate
  
      e^-itP_cφ_L^r,2≲ |t|^-1/2(1-2/r)φ_L^r',2,   2≤ r<∞,
   
  whereL^a,bdenotes the Lorentz space (see Section <ref>).
  
  We may also interpolate (<ref>) with (<ref>), leading to 
  
      ⟨ x⟩^-(1-2/r)e^-itP_cφ_L^r,2≲ |t|^-3/2(1-2/r)⟨ x⟩^1-2/rφ_L^r',2,   2≤ r<∞.
  
  In particular, interpolating between these last two estimates yields
  
      ⟨ x⟩^-θ(1-2/r)e^-itP_cφ_L^r,2≲ |t|^-(1/2+θ)(1-2/r)⟨ x⟩^θ(1-2/r)φ_L^r',2
  
  for any2≤ r<∞andθ∈[0,1].  Using this estimate together with Hölder's inequality, we can establish the following weightedL^2estimate:
   For any 2≤ r<∞ and α∈[12-1r,32-3r],  
  
      ⟨ x⟩^-αe^-itP_c φ_L^2≲ |t|^-α⟨ x⟩^α-[1/2-1/r]φ_L^r',2.
   Using Hölder's inequality (for Lorentz spaces) and (<ref>), we estimate
  
      ⟨ x⟩^-αe^-itP_c φ_L^2   ≲⟨ x⟩^-[1/2-1/r]_L^2r/r-2,∞⟨ x⟩^-α+[1/2-1/r]e^-itP_cφ_L^r,2
         ≲ |t|^-(1-2/r)(1/2+β)⟨ x⟩^α-[1/2-1/r]φ_L^r',2,
  
  where β∈[0,1] is determined by imposing
  
      β(1-2r)=α-[12-1r].
  
  In particular, the requirement that β belong to [0,1] requires α∈[12-1r,32-3r].  Noting that
  
      (1-2r)(12+β)=α
  
  then completes the proof. 
  
  
  
  
  
  
  
  § PROOF OF ASYMPTOTIC STABILITY
  
  
  In this section, we prove the main result (Theorem <ref>). We fixq<0and fixω_0such thatQ_ω_0is orbitally stable andω_0obeys the spectral condition appearing in Definition <ref>.  Let us also fix a weight1<α<min{p/4,32}, which will appear in the bootstrap norm below (see (<ref>)).  In (<ref>) and in the arguments below, we use notation such as∂_ω Q_ω(t)or∂_ω Q_ωas short-hand for the expression∂_ω Q_ω|_ω=ω(t).  
  
  By the orbital stability results of <cit.>, we know that any solution that starts close toQ_ω_0inH^1will be global and remain close to the orbit ofQ_ω_0inH^1for all time.  We begin by upgrading the orbital stability to a decomposition that incorporates the orthogonality conditions needed to establish decay and scattering for the perturbative part of the solution.  
  
   There exists =(ω_0) sufficiently small and η=η() sufficiently small that if
  
      u_0=Q_ω_0+ṽ_0withṽ_0_H^1<η
  
  and u(t) is the corresponding solution to (<ref>), then there exists a decomposition
  
      u(t) = e^iΦ(t){Q_ω(t) + v(t)}for allt≥ 0
  
  obeying the following properties:
  
    *  The phase has the following form: 
  
      Φ(t):=θ(t)+∫_0^t ω(s) ds,
  
  and ω(t) obeys 
  
      sup_t∈[0,∞)|ω(t)-ω_0|<.
  
    *  The following orthogonality conditions hold for v(t): 
  
      ⟨ v(t),Q_ω(t)⟩ = ⟨ v(t),i∂_ω Q_ω(t)⟩ = 0.
  
    *  The following estimates hold for v(t):
  
      ⟨ x⟩^α v(0)_L^2   ≤⟨ x⟩^αṽ_0_L^2 + 𝒪_ω_0(), 
      
      sup_t∈[0,∞)v(t)_H^1   ≲_ω_0.
  
    *  The following evolution equation holds for v(t):
  
      i∂_t v -θ̇v = (H+ω)v + e^-iΦf(u)-f(Q_ω) + θ̇Q_ω - iω̇∂_ω Q_ω.
    We let >0 be a small parameter to be determined below, and take ν=ν()>0 and δ=δ(ν)>0 to be determined below as well.  By orbital stability of Q_ω_0, we may find η=η(δ)>0 and β(t)∈ (with β(0)=0) such that
  
      ṽ_0_H^1<ηsup_t≥ 0u(t)-e^iβ(t)Q_ω_0_H^1<δ.
  
  
  We will construct the modulation parameters by the implicit function theorem.  We begin by defining the function F:L^2×^2→^2 by 
  
      F(h,θ,ω) = [[ F_1(h,θ,ω); F_2(h,θ,ω) ]]     = [[      ⟨ e^-iθh-Q_ω,Q_ω⟩; ⟨ e^-iθh-Q_ω,i∂_ω Q_ω⟩ ]] 
          = [[ ⟨ e^-iθh,Q_ω⟩ - Q_ω_L^2^2;        ⟨ e^-iθh,i∂_ω Q_ω⟩ ]].
  
  By construction, we have
  
      F(e^iβ(t)Q_ω_0,β(t),ω_0) ≡ 0.
  
  Next, by direct computation, 
  
      [[ ∂_θ F_1 ∂_ω F_1; ∂_θ F_2 ∂_ω F_2 ]] = [[                     ⟨ -ie^-iθh,Q_ω⟩ ⟨ e^-iθh,∂_ω Q_ω⟩ - 2⟨ Q_ω,∂_ω Q_ω⟩;                ⟨ -ie^-iθh,i∂_ω Q_ω⟩                ⟨ e^-iθh,i∂_ω^2 Q_ω⟩ ]].
  
  In particular, evaluating at (e^iβ(t)Q_ω_0,β(t),ω_0) yields the matrix 
  
      J = [[                   0 -⟨ Q_ω_0,∂_ω Q_ω_0⟩; -⟨ Q_ω_0,∂_ω Q_ω_0⟩                   0 ]],
  
  which is boundedly invertible (with norm depending on ω_0).  Note that the matrix J and the bounds on the derivatives of F above are independent of t.  This will allow us to apply the implicit function theorem to the entire curve of zeros {(e^iβ(t)Q_ω_0,β(t),ω_0):t≥ 0}, yielding neighborhoods around e^iβ(t)Q_ω and (β(t),ω_0) of fixed size (that is, with size independent of t). In particular, by the implicit function theorem, if we choose ν=ν(ω_0) and δ=δ(ν) sufficiently small, we may find for each t≥ 0 a map
  
      T_t:B_δ(e^iβ(t)Q_ω)⊂ L^2→ B_ν((β(t),ω_0)) ⊂^2
  
  such that
  
      F(h,T_t(h))=0 for allh∈ B_δ(e^iβ(t)Q_ω_0).
  
  As (<ref>) implies u(t)∈ B_δ(e^iβ(t)Q_ω_0) for all t≥ 0, we may choose
  
      (Φ(t),ω(t)):=T_t(u(t))
  
  and set
  
      v(t):=e^-iΦ(t)u(t)-Q_ω(t),  θ(t):=Φ(t)-∫_0^tω(s) ds,
   
  to arrive at the desired decomposition with (<ref>) and (<ref>).  We also have (<ref>) provided we take ν≤. 
  
  We next establish (<ref>). We first write
  
      Q_ω_0+ṽ_0 = u_0 = e^iθ(0){Q_ω(0)+v(0)},
  
  so that
  
      ⟨ x⟩^α v(0)_L^2≤⟨ x⟩^αṽ_0_L^2 + ⟨ x⟩^α[Q_ω_0-e^iθ(0)Q_ω(0)]_L^2.
  
  Now we estimate
  
      ⟨ x⟩^α[Q_ω_0-e^iθ(0)Q_ω(0)]_L^2
         ≤ |θ(0)|·⟨ x⟩^α Q_ω_0_L^2 + ⟨ x⟩^α[Q_ω_0-Q_ω(0)]_L^2.
  
  By construction, we have |θ(0)|<ν≤ (cf. β(0)=0).  Similarly, as |ω(0)-ω_0|<ν, we may choose ν=ν(,ω_0) possibly even smaller to guarantee that the second term above is bounded by .  Continuing from above, we derive (<ref>).
  
  The proof of (<ref>) follows in a similar fashion.  We first write
  
      e^iΦ(t){Q_ω(t)+v(t)} = e^iβ(t)Q_ω_0+ [u(t) - e^iβ(t)Q_ω_0],
  
  so that by (<ref>) we have
  
      v(t)_H^1≤e^iΦ(t)Q_ω(t) - e^iβ(t)Q_ω_0_H^1+δ.
  
  We now observe that
  
      e^iΦ(t)Q_ω(t) - e^iβ(t)Q_ω_0_H^1≤ |Φ(t)-β(t)|·Q_ω_0_H^1 + Q_ω(t)-Q_ω_0_H^1.
  
  Once again we have |Φ(t)-β(t)|< ν≤, and as |ω(t)-ω_0|<ν, we may choose ν=ν(,ω_0) possibly even smaller we may guarantee that the second term above is bounded by . This proves (<ref>).
  
  Finally, (<ref>) follows from (<ref>) and (<ref>). 
  
  
  We will show that with the decomposition (<ref>), the perturbationv(t)in fact converges to zero in suitable norms, and the modulation parameterω(t)converges ast→∞.
  
  
  
  By differentiating the orthogonality relations (<ref>), we first obtain an ODE system describing the evolution of the modulation parameters(ω,θ).  The choice of orthogonality conditions guarantees that the nonlinearity in the equation below contains terms that are quadratic and higher inv. 
  
   The modulation parameters (ω,θ) obey the ODE system
  
      A[[ ω̇; θ̇ ]] = [[      ⟨,Q_ω⟩; -⟨,∂_ω Q_ω⟩ ]],
  
  where A=A(t) is defined by 
  
      A:=[[      ⟨ Q_ω,∂_ω Q⟩-⟨ v,∂_ω Q⟩                    -⟨ v,Q_ω⟩;              -⟨ v,∂_ω^2 Q_ω⟩ ⟨ Q_ω,∂_ω Q_ω⟩+ ⟨ v,∂_ω Q_ω⟩ ]]
  
  and
  
      := e^-iΦ f(u)-f(Q_ω)-σp+22Q_ω^p v - σp2Q_ω^p v̅.
   We start with (<ref>), multiply by Q_ω, integrate, and take the imaginary part.  Using self-adjointness of H+ω, this yields
  
      ⟨v̇,Q_ω⟩
          = θ̇⟨ v,Q_ω⟩ -ω̇⟨ Q_ω,∂_ω Q_ω⟩ +⟨ e^-iΦ f(u)-f(Q_ω),Q_ω⟩-⟨ v,f(Q_ω)⟩.
  
  On the other hand, differentiating the first equation in (<ref>) implies
  
      [⟨v̇,Q_ω⟩ + ω̇⟨ v,∂_ω Q_ω⟩] =0,
  
  so that continuing from above yields
  
      ω̇[⟨ Q_ω,∂_ω    Q_ω⟩-⟨ v,∂_ω  Q_ω⟩] -θ̇⟨ v,Q_ω⟩
         = ⟨ e^-iΦ f(u)-f(Q_ω),Q_ω⟩-⟨ v,f(Q_ω)⟩  = ⟨,Q_ω⟩,
  
  with  defined as in (<ref>). 
  
  Similarly, we start with (<ref>), multiply by i∂_ω Q, integrate, and take the imaginary part.  This yields 
  
      -⟨v̇,∂_ω Q_ω⟩    = θ̇⟨ v,∂_ω Q_ω⟩ + θ̇⟨ Q_ω,∂_ω Q_ω⟩
            + ⟨ e^-iΦ f(u)-f(Q_ω),∂_ω Q_ω⟩ + ⟨ v,(H+ω)∂_ω Q_ω⟩.
  
  Recalling (<ref>), we have
  
      (H+ω)∂_ω Q_ω = - Q_ω - σ(p+1)Q_ω^p ∂_ω Q_ω,
  
  so that using (<ref>), we arrive at
  
      ⟨ v,(H+ω)∂_ω Q_ω⟩ =  - σ(p+1)⟨ v,Q_ω^p∂_ω Q_ω⟩.
  
  On the other hand, differentiating the second equation in (<ref>) yields 
  
      ⟨v̇,∂_ω Q_ω⟩ + ω̇⟨ v,∂_ω^2 Q_ω⟩ = 0.
  
  Thus we arrive at
  
      ω̇⟨ v,∂_ω^2     Q_ω⟩ - θ̇[⟨ v,∂_ω Q_ω⟩ + ⟨ Q_ω,∂_ω Q_ω⟩] 
          = ⟨ e^-iΦ f(u)-f(Q_ω),∂_ω Q_ω⟩ -σ(p+1)⟨ v,Q_ω^p∂_ω Q_ω⟩
          = ⟨,∂_ω Q_ω⟩.
  
  
  Collecting (<ref>) and (<ref>) yields (<ref>). 
  
  
  We turn to the problem of proving global-in-time estimates forv(t)by a bootstrap argument.  We will proceed by proving estimates on intervals of the form[0,T].  It is difficult to analyzev(t)directly due to the fact that the linear operator in (<ref>) is time-dependent.  Instead, givenT>0, we will introduce a new variable that will solve an equation with a time-independent linear part.
  
   Given T>0, we set
  
      Ψ(t) = θ(T) + tω(T)
  
  and define z(t) via
  
      e^iΦ(t)v(t) = e^iΨ(t)z(t).
  
  Then z(t) obeys the evolution equation
  
      i∂_t z = [H+ω(T)]z + σp+22Q_ω(T)^p z + σp2Q_ω(T)^p z̅ + ,
  
  where
  
      ℱ    = e^iΦ-iΨ{+ θ̇Q_ω - iω̇∂_ω Q_ω}
            + σp+22[Q_ω^p-Q_ω(T)^p]z 
            + σp2[Q_ω^p - Q_ω(T)^p]z̅ + σp2[e^2iΦ-2iΨ-1]Q_ω^p z̅.
   To derive (<ref>), we first use (<ref>) along with
  
      Φ̇(t) = θ̇(t)+ω(t)andΨ̇(t)=ω(T)
  
  to derive
  
      i∂_t z = [H+ω(T)]z + e^iΦ-iΨ{ e^-iΦf(u)-f(Q)+θ̇Q_ω - iω̇∂_ω Q_ω}.
  
  After isolating the parts of e^-iΦf(u)-f(Q) that are linear in v and writing 
  
      e^iΦ-iΨ   {σp+22Q_ω^p v + σp2Q_ω^p v̅}
          = σp+22Q_ω(T)^p z + σp+22[Q_ω^p-Q_ω(T)^p]z 
            + σp2Q_ω(T)^p z̅ + σp2[Q_ω^p - Q_ω(T)^p]z̅ + σp2[e^2iΦ-2iΨ-1]Q_ω^p z̅,
  
  we arrive at (<ref>). 
  
  
  The introduction ofz(t)solves one problem (that is, the issue of the time-dependent linear operator) but leads to a new one.  In particular,zdoes not obey the orthogonality relation needed to apply the dispersive estimates adapted to the linear operator appearing in (<ref>).  To remedy this, we introduce one more variable. For this, we recall Proposition <ref> and introduce the notationP_c(T)for the projection onto the continuous spectrum ofŁ(ω(T)). 
  
   For each t, we have a spectral decomposition
  
      z(t)=ia(t)Q_ω(T)+b(t)∂_ω Q_ω(T)+ η(t),
  
  so that η(t)=P_c(T)z(t). The coefficients (a,b) obey the linear system
  
      B[[ a; b ]] = -[[       ⟨ e^i[Ψ-Φ]η,Q_ω(t); ⟨ e^i[Ψ-Φ]η,i∂_ω Q_ω(t)⟩ ]],
  
  where 
  
      B=[[      -sin(Ψ-Φ) ⟨ Q_ω(T),Q_ω(t)⟩     cos(Ψ-Φ)⟨∂_ω Q_ω(T),Q_ω(t)⟩;   cos(Ψ-Φ) ⟨ Q_ω(T),∂_ω Q_ω(t)⟩ sin(Ψ-Φ)⟨∂_ω Q_ω(T),∂_ω Q_ω(t)⟩ ]].
  
  The function η (identified with the vector [η η]^t) obeys the integral equation
  
      η(t) = e^tŁ(ω(T))P_c(T)z(0) + P_c(T)∫_0^t e^(t-s)Ł(ω(T))[[  ; - ]] ds.
   The system (<ref>) with (<ref>) follows from direct computation using (<ref>).  
  
  As for (<ref>), we first re-cast (<ref>) as the following system for (z_1,z_2): 
  
      ∂_t[[ z_1; z_2 ]] = Ł(ω(T))[[ z_1; z_2 ]] + [[  ; - ]],
  
  where Ł is as in (<ref>). We now apply P_c(T) and recall that P_c(T)iQ_ω(T)=P_c(T)∂_ω Q_ω(T)=0, yielding an evolution equation for η.  Recasting this equation in integral form yields (<ref>).  
  
  
  We now turn to the bootstrap argument.   We fixT>0and first introduce the quantities that we will need to control.  We recall that we have chosen a weightαobeying
  
      1<α<min{p4,32},
  
  which requiresp>4. We then fix a large but finite exponentr=r(p,α), which will be determined more precisely below (see Lemma <ref> and its proof).  We define
  
  
      v_X([0,T])    = sup_t∈[0,T]{|t|^1/2-1/rv(t)_L_x^r,2 + |t|^-α⟨ x⟩^-αv(t)_L_x^2}, 
      (ω,θ)_Y([0,T])    = sup_t∈[0,T]{⟨ t⟩^2α[ |ω̇(t)|+|θ̇(t)|]}, 
      Q_ω_Z([0,T])    =sup_t∈[0,T]{⟨ x⟩^20 Q_ω_H^1+⟨ x⟩^20∂_ω Q_ω_L^2 + ⟨ x⟩^20∂_ω^2 Q_ω_L^2},
  
  whereL^r,2denotes the Lorentz space. The weight⟨ x⟩^20does not have any special significance; any `sufficiently large' weight would do.
  
  We first collect some bounds that depend only onω_0. First, recall that⟨ Q_ω_0,∂_ω Q_ω_0⟩≠ 0(see e.g. Section <ref>).  Thus, choosing=(ω_0)possibly even smaller in Proposition <ref>, we can guarantee (via (<ref>)) that 
  
      inf_t∈[0,∞]|⟨ Q_ω,∂_ω Q_ω|⟩≳_ω_0 1.
  
  Next, noting that
   
      ⟨ x⟩^20Q_ω_0_H^1 + ⟨ x⟩^20∂_ω Q_ω_0_L^2 + ⟨ x⟩^20∂_ω^2 Q_ω_0_L^2≲_ω_0 1
  
  we can choose=(ω_0)possibly even smaller and  guarantee via (<ref>) that
  
      Q_ω_Z([0,∞])≲_ω_01
  
  (see Section <ref>). 
  
  We suppose that the following estimates hold on[0,T]:
  
      v_X([0,T])   ≤ 2C_1, 
      (ω,θ)_Y([0,T])   ≤ 2C_2^2,
  
  where>0is as in Proposition <ref> and the constantsC_1,C_2will be determined below; in particular, we will need to chooseC_1depending onω_0andC_2dependingC_1. We will also need to takepossibly even smaller depending on the constants{C_j}_j=1^2and the powerp.  The parameters we choose will also depend onq(the coupling constant for the delta potential); however, asqis fixed throughout the paper, we will not track this dependence throughout the argument. 
  
  Henceforth we assume that the estimates (<ref>) and (<ref>) hold.  We will show we may in fact improve the estimates (<ref>) and (<ref>) on[0,T], removing the factor2in each estimate.  Using this and the smallness assumption at timet=0, a standard continuity argument will imply that in fact, (<ref>) and (<ref>) hold on the entire interval[0,∞). 
  
  We first record a weighted estimate forv, which follows from a straightforward virial estimate at the level of the original solutionu. Because the estimate involves the weight|x|^2, the delta potential ultimately has no real effect (e.g. when integrating by parts).  We will proceed with the formal computation and refer the reader to <cit.> for a more detailed and careful presentation.
  
  
   We have
  
      ⟨ x⟩ v(t)_L^2≲_ω_0⟨ t⟩
  
  uniformly in t∈[0,T].
   By orbital stability, we have
  
      u_L_t^∞Ḣ_x^1([0,T]×^3)≲_ω_0 1.
  
  We now use the virial identity.  By direct computation using (<ref>) and integration by parts, we have
  
      | ddtxu(t)_L^2^2| =| 4∫ xu̅∂_x u dx| ≤ 4xu(t)_L^2∂_x u(t)_L^2,
  
  which implies
  
      xu(t)_L^2≤xu_0_L^2+2tu_L_t^∞Ḣ_x^1.
  
  The result now follows from the triangle inequality.  
  
  The next estimate provides control over the modulation parameters.
  
   If  is sufficiently small, the matrix A appearing in (<ref>) is invertible for all t∈[0,T], with
  
      A^-1≲_ω_0,C_1 1
  
  uniformly in t∈[0,1].  The modulation parameters (ω,θ) obey 
  
      (ω,θ)_Y([0,T])≲_p,ω_0,C_1v_X([0,T])^2.
  
  In particular, by (<ref>), 
  
      (ω,θ)_Y([0,T])≲_p,ω_0,C_1^2.
   Using (<ref>) and (<ref>), we compute
  
      A   ≥⟨ Q_ω,∂_ω Q_ω⟩^2 - |⟨ v,∂_ω Q_ω⟩|^2 - |⟨ v,Q_ω⟩| |⟨ v,∂_ω^2 Q_ω⟩| 
         ≳_ω_0 1 - 𝒪_ω_0, C_1(^2)≳_ω_0 1
  
  for =(ω_0, C_1) sufficiently small.  Using this, we may deduce (<ref>).
  
  We turn to (<ref>) (which then implies (<ref>) via (<ref>)). In light of (<ref>) and (<ref>), it is enough to prove
  
      ⟨ x⟩^-2α_L^1≲_p,ω⟨ t⟩^-2αv_X^2,
  
  uniformly in t∈[0,1], where  is as in (<ref>). For this we observe that by Young's inequality, Sobolev embedding, and (<ref>), we have the pointwise estimate
  
      || ≲_p |v|^2|Q_ω|^p-1+|v|^p+1≲_ω_0 |v|^2.
  
  Now (<ref>) follows from the fact that 
  
      ⟨ x⟩^-α v(t)_L^2≲_C_1⟨ t⟩^-αv_X([0,T])
  
  uniformly in t∈[0,T]. 
  
  
  We next estimate the spectral parametersa(t)andb(t), which in turn allows us to controlv(t)(usingη(t)=P_c(T)z(t)). 
  
   If  is sufficiently small, matrix B appearing in (<ref>) is invertible for all t∈[0,T], with
  
      B^-1≲_ω_0 1.
  
  The coefficients (a,b) obey
  
      sup_t∈[0,T]⟨ t⟩^α{|a(t)| + |b(t)|}≲_ω_0η_X([0,T]),
  
  and consequently
  
      v_X([0,T])≲_ω_0η_X([0,T]).
   We first abbreviate s^2 = sin^2(Ψ-Φ) and c^2=cos^2(Ψ-Φ) and compute
  
      - B     = s^2⟨ Q_ω(T),Q_ω⟩⟨∂_ω Q_ω(T),∂_ω Q_ω⟩ + c^2⟨∂_ω Q_ω(T),Q_ω⟩⟨ Q_ω(T),∂_ω Q_ω⟩
          = ⟨∂_ω Q_ω(T),Q_ω⟩⟨ Q_ω(T),∂_ω Q_ω⟩
             + s^2[  ⟨ Q_ω(T),Q_ω⟩⟨∂_ω Q_ω(T),∂_ω Q_ω⟩  - ⟨∂_ω Q_ω(T),Q_ω⟩⟨ Q_ω(T),∂_ω Q_ω⟩].
  
  For (<ref>), we begin by writing
  
      ⟨   ∂_ω Q_ω(T),Q_ω⟩⟨ Q_ω(T),∂_ω Q_ω⟩
         ≥ [⟨∂_ω Q_ω,Q_ω⟩]^2  - 𝒪_ω_0( Q_ω - Q_ω(T)_L^2 + ∂_ω Q_ω - ∂_ω Q_ω(T)_L^2).
  
  We now observe that by (<ref>), 
  
      Q_ω -Q_ω(T)_L^2≲_ω_0∫_t^T |ω̇(s)| ds ≲_ω_0,C_2∫_t^T ⟨ s⟩^-2α ds≲_ω_0,C_2^2
  
  uniformly in t∈[0,T], with a similar bound for ∂_ω Q_ω - ∂_ω Q_ω(T)_L^2.  In particular, using (<ref>), we obtain
  
      (<ref>)≳_ω_0 1 - 𝒪_ω_0,C_2(^2) ≳_ω_01
  
  for =(ω_0,C_2) sufficiently small. 
  
  It remains to estimate (<ref>).  We begin with
  
      |(<ref>)| ≲_ω_0 |Ψ-Φ|^2.
  
  We now claim that
  
      sup_t∈[0,T]|Ψ(t)-Φ(t)| ≲_C_2^2,
  
  so that for  chosen possibly even smaller, we may continue from above to deduce
  
      | B|≳_ω_0 1
  
  uniformly over t∈[0,T].  We begin by writing
  
      Φ(t)-Ψ(t)=θ(t)-θ(T)+∫_0^t [ω(s)-ω(T)] ds.
  
  Using (<ref>), we then have
  
      |θ(t)-θ(T)|≤∫_t^T |θ̇(s)| ds ≲_C_2^2∫_t^T ⟨ s⟩^-2α ds ≲_C_2^2.
  
  Similarly, recalling α>1, 
  
      ∫_0^t |ω(s)-ω(T)| ds    ≤∫_0^t ∫_s^T |ω̇(τ)| dτ ds 
         ≲_C_2^2∫_0^t∫_s^T ⟨τ⟩^-2α dτ ds ≲_C_2^2,
  
  thus completing the proof of (<ref>).  
  
  In particular, we have now proven (<ref>), which shows B(t) is invertible for all t∈[0,T].  Using (<ref>) as well, we readily deduce (<ref>). 
  
  We turn to (<ref>).  In fact, this follows from (<ref>), (<ref>), (<ref>), and the straightforward estimate
  
      |⟨ |η|,|Q_ω|⟩| ≤⟨ x⟩^-αη_L^2⟨ x⟩^α Q_ω_L^2,
  
  with a similar estimate using ∂_ω Q_ω in place of Q_ω.  Finally, recalling (<ref>) and (<ref>), we see that (<ref>) implies (<ref>).
  
  
  
  We turn to the estimation ofη.  In light of (<ref>), control overηwill yield control overv.  We will use the evolution equation (<ref>), relying on the dispersive estimates for the linearized operator.  We begin by recording a pointwise-in-time bound for the nonlinearity(see (<ref>)).
  
   For r=r(α,p) sufficiently large, 
  
      ⟨ x⟩^α-[1/2-1/r]_L^r',2+_L^2≲_p,ω_0,C_1,C_2⟨ t⟩^-α^2,
  
  uniformly over t∈[0,T], where 1r+1r'=1. 
   We begin by observing that in light of (<ref>), we may upgrade the bootstrap assumptions
  
      v(t)_L^r,2≲_C_1 |t|^-(1/2-1/r)and⟨ x⟩^-αv(t)_L^2≲_C_1 |t|^-α
  
  to
  
      v(t)_L^r,2≲_ω_0,C_1⟨ t⟩^-(1/2-1/r)and⟨ x⟩^-αv(t)_L^2≲_C_1⟨ t⟩^-α.
  
  
  We will first estimate the weighted norm.  For this, we first estimate the contribution of  to  (see (<ref>)). We rely on the pointwise estimate
  
      || ≲_p |Q_ω|^p-1|v|^2 + |v|^p+1.
  
  
  We begin with the purely nonlinear term |v|^p+1. We define the parameter
  
      k=2(r-(p+2))r-2,
  
  where we impose r>2(p+1) to guarantee that 1<k<2. 
  
  We may then estimate
  
      ⟨ x⟩^α-[1/2-1/r]|v|^p+1_L^r',2   ≲⟨ x⟩^α-[1/2-1/r]v^k _L^2/k,2/k-1v^p+1-k_L^r/p+1-k,2/2-k
         ≲⟨ x⟩^1/k(α-[1/2-1/r])v_L^2^k v_L^r,2/2-k(p+1-k)^p+1-k,
  
  where we have used 2k/k-1>2.  For the weighted term, we will use Lemma <ref>.  This requires the condition
  
      1k(α-[12-1r]) ≤ 1.
  
  Recalling the definition of k and rearranging, we find that this follows provided
  
      α≤52-1r-2pr-2.
  
  In particular, as α<32, we can guarantee this condition by choosing r sufficiently large.  We will control the remaining norm in (<ref>) by the L^r,2-norm.  This requires that
  
      22-k(p+1-k)>2,
  
  which after rearranging is equivalent to p>1.  In particular, we arrive at the estimate
  
      ⟨ x⟩^α-[1/2-1/r]|v|^p+1_L^r',2≲^p+1-k⟨ t⟩^α-[1/2-1/r]-(p+1-k)(1/2-1/r).
  
  We now observe that the condition
  
      α-[12-1r]-(p+1-k)(12-1r)<-α
  
  follows from α<p4. We also note that p+1-k>2 follows from k<2<p-1.  Thus we have
  
      ⟨ x⟩^α-[1/2-1/r]|v|^p+1_L^r',2≲^2 ⟨ t⟩^-α,
  
  as desired.
  
  Next, we estimate
  
      ⟨ x⟩^α-[1/2-1/r] Q_ω^p-1v^2 _L^r',2   ≲⟨ x⟩^-r-3/r-2αv_L^2r',4^2 ⟨ x⟩^α-[1/2-1/r]+2(r-3)/r-2αQ_ω^p-1_L^∞
         ≲⟨ x⟩^-r-3/r-2αv_L^2r',2^2 Q_ω_Z^p-1.
  
  To arrive at a suitable estimate, it therefore remains to show that
  
      ⟨ x⟩^-r-3/r-2αv_L^2r',2≲⟨ t⟩^-α/2.
  
  We first have the following estimate by interpolation: 
  
      ⟨ x⟩^-r-3/r-2αv_L^2r',2   ≲⟨ x⟩^-αv _L^2^r-3/r-2v_L^r,2^1/r-2
         ≲⟨ t⟩^-r-3/r-2α -1/r-2(1/2-1/r).
  
  In particular, we need to show that 
  
      -r-3r-2α -1r-2(12-1r)<-α2.
  
  In fact, the left-hand side converges to -α as r→∞, and so this holds for any large r.
  
  Collecting the estimates above, this completes the estimation of the L^r',2-norm of .
  
  We turn to the contribution of the modulation parameters to . By (<ref>), (<ref>), and Hölder's inequality, we have
  
      ⟨ x⟩^α-[1/2-1/r]   θ̇Q_ω_L^r',2 + ⟨ x⟩^α-[1/2-1/r]ω̇∂_ω Q_ω_L^r',2
         ≲_C_2⟨ t⟩^-2α^2Q_ω_Z ≲_ω_0,C_2⟨ t⟩^-2α^2,
  
  which is acceptable. 
  
  We next use (<ref>), (<ref>), and (<ref>) to estimate
  
      ⟨    x⟩^α-[1/2-1/r][Q_ω^p-Q_ω(T)^p]z_L^r',2
         ≲⟨ x⟩^-αz_L^2⟨ x⟩^4[Q_ω^p-Q_ω(T)^p]_L^2r/r-2
         ≲_C_1,p⟨ t⟩^-α⟨ x⟩^4  ∫_t^T ω̇(s)· Q_ω(s)^p-1∂_ω Q_ω(s) ds_H_x^1
         ≲_C_1,C_2,p^3⟨ t⟩^-α∫_t^T ⟨ s⟩^-2α⟨ x⟩^4 Q_ω(s)^p-1∂_ω Q_ω(s)_H_x^1 ds 
         ≲_ω_0,C_1,C_2,p^3⟨ t⟩^-3α+1,
  
  which is acceptable. 
  
  Finally, we use (<ref>), (<ref>), and (<ref>) to estimate
  
      ⟨ x⟩^α-[1/2-1/r]|e^2iΦ-2iΨ-1]Q_ω^p z _L^r',2   ≲ |Φ-Ψ|·⟨ x⟩^-αz(t)_L^2⟨ x⟩^4 Q_ω^p_L^2r/r-2
         ≲_ω_0,C_1,C_2,p^3 ⟨ t⟩^-α⟨ x⟩^4 Q_ω^p_H^1
         ≲_ω_0,C_1,C_2^3⟨ t⟩^-α,
  
  which is acceptable.  This completes the estimate of the weighted norm.
  
  It remains to estimate  in L^2. Again, we begin with the contribution of , relying on the pointwise estimate (<ref>).  We begin by estimating
  
      |v|^p+1_L^2≲v_L^2(p+1)^p+1≲^p+1⟨ t⟩^-p/2,
  
  where we have used interpolation between the L^2 and L^r norms of v (and assumed r>2(p+1)).  As α<p/2, the contribution of this term is acceptable.  On the other hand, by interpolation,
  
      v^2 Q_ω^p-1_L^2   ≲⟨ x⟩^-α[r-4/2(r-2)]v_L^4^2 ⟨ x⟩^4 Q_ω^p-1_L^∞
         ≲_ω_0⟨ x⟩^-αv_L^2^r-4/r-2v_L^r^r/r-2
         ≲_ω_0^2 ⟨ t⟩^-α[r-4/r-2]-[1/2-1/r][r/r-2].
  
  As the power tends to -α-1/2 as r→∞, we find that this contribution is acceptable provided r is sufficiently large. 
  
  It remains to estimate the contribution of the modulation parameters. First,
  
      θ̇Q_ω_L^2+ω̇∂_ω Q_ω_L^2≲_C_2^2⟨ t⟩^-2αQ_ω_Z≲_ω_0,C_2^2⟨ t⟩^-2α,
  
  which is acceptable. Next, estimating as we did above, we have
  
      [Q_ω^p - Q_ω(T)^p]z_L^2   ≲⟨ x⟩^-αz_L^2∫_t^T ⟨ s⟩^-2αQ_ω(s)^p-1∂_ω Q_ω(s)_H^1 ds 
         ≲_ω_0,C_1,C_2,p^3⟨ t⟩^-3α+1,
  
  which is acceptable. Finally, estimating again as we did above,
  
      [e^2iΦ-2iΨ-1]Q_ω^p z_L^2≲ |Φ-Ψ|·⟨ x⟩^-αz_L^2Q_ω_Z^p ≲_ω_0,C_1,C_2^3⟨ t⟩^-α.
  
  This completes the proof.  
  
  
  We can now establish our desired estimate forη. 
  
   We have the following estimate for η:
  
      η_X([0,T])≲_ω_0⟨ x⟩^α v(0)_L^2+ 𝒪_p,ω_0,C_1,C_2(^2)
   We will utilize the Duhamel formula in (<ref>) and apply dispersive estimates adapted to e^tŁ(ω(T))P_c(T) (see (<ref>) and (<ref>)).  By (<ref>), we may guarantee that the implicit constants appearing in the dispersive estimates for all ω(t) are all bounded by a constant depending only on ω_0.  Also, as α<32 by assumption, we may guarantee α<32-3r (which is necessary to apply (<ref>)) provided r is chosen sufficiently large. 
  
  We first consider the L^r,2-norm.  We apply (<ref>), Hölder's inequality, and Lemma <ref> to obtain
  
      |t|^1/2-1/rη(t) _L^r,2   ≲_ω_0z(0)_L^r',2 + |t|^1/2-1/r∫_0^t |t-s|^-[1/2-1/r](s)_L^r',2 ds 
         ≲_ω_0⟨ x⟩^1/2-1/rz(0)_L^2 +^2 |t|^1/2-1/r∫_0^t |t-s|^-[1/2-1/r]⟨ s⟩^-α ds 
         ≲_ω_0⟨ x⟩^αv(0)_L^2 + 𝒪_p,ω_0,C_1,C_2(^2)
  
  uniformly over t∈[0,T], which is acceptable.  For the weighted norm, we use both (<ref>) and (<ref>) to arrive at the estimate
  
      |t|^α   ⟨ x⟩^-αη(t)_L^2≲_ω_0⟨ x⟩^α-[1/2-1/r]z(0)_L^r',2
            + |t|^α∫_0^t min{|t-s|^-α,1}{⟨ x⟩^α-[1/2-1/r]_L^r',2+_L^2} ds 
         ≲_ω_0⟨ x⟩^αv(0)_L^2 + ^2|t|^α∫_0^t min{|t-s|^-α,1}⟨ s⟩^-α ds 
         ≲_ω_0⟨ x⟩^αv(0)_L^2 + 𝒪_p,ω_0,C_1,C_2(^2)
  
  uniformly over t∈[0,T], which is acceptable.  This completes the proof.
  
  
  With the preceding lemmas in place, we can complete the bootstrap argument.
  
   There exist C_1,C_2>0 depending on ω_0 and p such that for =(ω_0,p)>0 sufficiently small and η=η()>0 sufficiently small, if
  
      ṽ_0_H^1+⟨ x⟩^αṽ_0_L^2≤η,
  
  then the estimates (<ref>) and (<ref>) hold uniformly for all T>0. 
   We first claim that the estimates (<ref>) and (<ref>) hold in a neighborhood of t=0.  We first consider the estimate (<ref>).  For this, we note that for small times we may control the X-norm by the L_t^∞ H_x^1-norm, which is 𝒪_ω_0() by (<ref>).  In particular, we need only choose C_1 sufficiently large depending on ω_0.  The estimate (<ref>) for small times follows from the estimate (<ref>) via (<ref>), choosing C_2 sufficiently large depending on p,ω_0,C_1. 
  
  Next, we suppose that the estimates (<ref>) and (<ref>) hold on an interval [0,T].  We will then show that for >0 sufficiently small, we have the improved estimates
  
      v_X([0,T])   ≤ C_1, 
      (ω,θ)_Y([0,T])   ≤ C_2^2.
  
  A standard continuity argument then implies the result. 
  
  Recall that for >0 small enough, the condition (<ref>) guarantees that the implicit constant in the dispersive estimates for Ł(ω(t)) is bounded uniformly in t. In particular, the implicit constant in Lemma <ref> is independent of T. 
  
  We first consider (<ref>).  As above, by (<ref>) we may guarantee that (<ref>) holds by choosing C_2 sufficiently large depending on the parameters p,ω_0, and  C_1. 
  
  Finally, we consider (<ref>).  We first apply Lemma <ref> and (<ref>), which yields
  
      η_X([0,T])≲_ω_0 + 𝒪_p,ω_0,C_1,C_2(^2).
  
  In particular, for =(p,ω_0,C_1,C_2) sufficiently small, we may guarantee that
  
      η_X([0,T)≲_ω_0.
  
  Now using (<ref>) and choosing C_1 sufficiently large depending on ω_0, we may derive (<ref>).  This completes the proof. 
  
  With the bootstrap closed and the global estimates forvand(ω,θ)in place, it is now straightfoward to prove thatv(t)scatters inL^2ast→∞. 
  
   There exists ω_+ (with |ω_0-ω_+|≲^2) and v_+∈ L^2 such that 
  
      lim_t→∞v(t) - e^tŁ(ω_+)v_+_L^2 = 0.
   We have the decomposition
  
      u(t) = e^iΦ(t){Q_ω(t)+v(t)},   Φ(t)=θ(t)+∫_0^t ω(s) ds,
  
  for all t≥ 0, with v and (θ,ω) obeying the estimates (<ref>) and (<ref>) on the entire interval [0,∞).  We now carry out a constriction on [0,∞) analogous to the one we used to prove estimates on [0,T].  In particular, we set
  
      θ_+ = lim_t→∞θ(t) andω_+ = lim_t→∞ω(t).
  
  Observe that by using the explicit decay rates obtained, we may also define the limit
  
      γ = ∫_0^∞ [ω(s)-ω_+] ds.
  
  
  We then define Ψ and z via 
  
      Ψ(t) = θ_+ + tω_+ and e^iΨ(t)z(t) = e^iΦ(t)v(t),
  
  and we formulate the equation solved by z with linear operator given by Ł(ω_+).  In particular, we obtain the PDE (<ref>) with (<ref>), with ω(T) replaced by ω_+.  We then introduce the variable η_+, with coefficients a_+ and b_+, as in (<ref>).  In particular η=P_c(ω_+)z and η evolves according to (<ref>), with ω(T) replaced by ω_+. 
  
  Now the estimates appearing in Lemma <ref> suffice to prove that e^-tℒ(ω_+)η(t) forms a Cauchy sequence in L^2 and hence has a limit η_+∈ L^2.  In light of the estimates on a_+ and b_+ (see (<ref>)), this implies z(t)=e^tŁ(ω_+)η_+ + o(1) as t→∞.  This in turn yields
  
      v(t) = e^i[Ψ_+(t)-Φ(t)]e^tŁ(ω_+)η_+ + o(1) = e^iγ e^tŁ(ω_+)η_+ + o(1)
  
  in L^2 as t→∞, where we have used the dominated convergence theorem as well.  The result now follows with v_+ = e^iγη_+. 
  
  Collecting the results of Propositions <ref>, <ref>, and <ref>, we have now completed the proof of Theorem <ref>. 
  
  
  
  
  § CONSTRUCTION OF ODE SOLUTIONS
  
  
  In this section we discuss the construction of the solutions to the generalized eigenvalue problem (<ref>) (f_1,f_2,f_3andf̃_4appearing in Lemma <ref>).  The construction is similar to what appears in <cit.>, with the main differences 
  arising from the presence of the jump condition atx=0. Accordingly, we shall be somewhat brief in our presentation.  
  
  We letH_q=-1/2_̣x^2 + q δ_0andℋ_V=ℋ_q + 𝒱, where
  
      ℋ_q := 
      	[  H_q + ω        0;        0 - H_q -ω ],   𝒱 :=
      	σ_3[ V_1(x) V_2(x); V_2(x) V_1(x) ].
  
  For now, we regardω>0just as a parameter.
  We let 
  
      μ = μ(ω):= √(ξ^2 + 4ω).
  
  In this section, the precise form of the potential𝒱appearing in the operator is not important.  We only assume thatV_j(x) ∈ C^(∖{0})are even, real-valued functions such that
  
      ∑_j=0^k (|_̣x^j V_1(x)| + |_̣x^j V_2(x)|) ≲_k e^-γ |x|
  
  holds for someγ>0and anyk≥ 0.  In our setting,V_1=σp+1/2Q_ω^pandV_2=σp/2Q_ω^p, so that the assumption above is satisfied; furthermore, the implicit constants (as well asγ) depend continuously onω.  (Recall that we needed to assert continuous dependence in Proposition <ref>.) 
  
  We consider the ODE system
  
      ℋ_V f =  ξ^2/2 + ωf ,    f(x;ξ) : ^2 →^2.
  
  By the standard ODE theory, one sees that the initial value problem of (<ref>) has solutions.  Our goal is to find four solutionsf_1,f_2,f_3, andf_4satisfying the asymptotic conditions
  
      f_1(x;ξ) → e^iξ x[ 1; 0 ], 
      	     f_2(x;ξ) → e^-iξ x[ 1; 0 ], 
         f_3(x;ξ) → e^-μ x[ 0; 1 ], 
           f_4(x;ξ) → e^μ x[ 0; 1 ]
  
  asx→.
  Before beginning the construction of our specific solutions, we collect some basic facts about the solution space.
  
  We first note that a solution to the homogeneous equation
  
      ℋ_q f=  12ξ^2 + ωf, 
      	   f(0) =[ a_1; a_2 ], 
      	   f'(0)=[ b_1; b_2 ]
  
  is given by
  
      f(x) = D_ξ' (x) 
      	[ a_1; a_2 ]
        + D_ξ (x) [ b_1; b_2 ]
      ,
  
  where
  
      D_ξ (y) = ∫_0^y [  cosξ s       0;       0 coshμ s ] ds = [  sinξ y/ξ         0;         0 sinhμ y/μ ],
  
  withμ=√(ξ^2 + 4ω)as above.
  
  Next, we have the following result.
  
  For each ξ∈,
  The solution space X(ξ) to (<ref>) is isomorphic to ^4.
  Moreover, for any fixed x_0 ∈∖{0}, the map
  
      X(ξ) ∋ f= [ f^(1)(x;ξ); f^(2)(x;ξ) ]↦[     f^(1)(x_0 ;ξ);      f^(2)(x_0;ξ); _̣x f^(1)(x_0 ;ξ);  _̣x f^(2)(x_0;ξ) ]∈^4
  
  is an isomorphism. Similarly,
  
      X(ξ) ∋ f= [ f^(1)(x;ξ); f^(2)(x;ξ) ]↦[      f^(1)(0 ;ξ);       f^(2)(0;ξ); _̣x f^(1)(0+ ;ξ);  _̣x f^(2)(0+;ξ) ]∈^4
  
  and
  
      X(ξ) ∋ f= [ f^(1)(x;ξ); f^(2)(x;ξ) ]↦[      f^(1)(0 ;ξ);       f^(2)(0;ξ); _̣x f^(1)(0- ;ξ);  _̣x f^(2)(0-;ξ) ]∈^4
  
  are also isomorphisms.  Here ± denote limits from the right/left.
   This is just a rephrasing of the existence of a unique solution for data given at a fixed time.  Thus it is very standard if we do not have the delta potential (that is, if q=0). Incorporating the delta potential requires that we add the jump condition
  
      [      f^(1)(0 ;ξ);       f^(2)(0;ξ); _̣x f^(1)(0+ ;ξ);  _̣x f^(2)(0+;ξ) ]
      	=
      	[  1  0  0  0;  0  1  0  0; 2q  0  1  0;  0 2q  0  1 ][      f^(1)(0 ;ξ);       f^(2)(0;ξ); _̣x f^(1)(0- ;ξ);  _̣x f^(2)(0-;ξ) ].
  
  As the matrix in the right hand side is invertible, the unique solvability follows by gluing together solutions in the positive and negative regions.
  
  Fix ξ∈. For solutions {f_j}_j∈ J⊂ X(ξ) to (<ref>), the following statements are equivalent
  
    *  The solutions {f_j}_j∈ J are linearly independent;
  
    *  For some x_0 ∈∖{0}, vectors
  	
      {[    f_j^(1) (x_0);     f_j^(2)(x_0); _̣x f_j^(1)(x_0); _̣x f_j^(2)(x_0) ]}_j∈ J⊂^4
  
  	are linearly independent;
  
    *  The statement (2) with x_0 replaced by 0-;
  
    *  The statement (2) with x_0 replaced by 0+.
  
  
  We next introduce a `scalar' Wronskian.
  
   Fix ξ∈.  For two solutions f_1, f_2 ∈ X(ξ), the Wronskian
  
      W[f_1,f_2] (x) := [     f_1^(1) (x);     f_1^(2) (x); _̣x f_1^(1) (x); _̣x f_1^(2) (x) ]^t
      	[  0  0 -1  0;  0  0  0  1;  1  0  0  0;  0 -1  0  0 ][     f_2^(1) (x);     f_2^(2) (x); _̣x f_2^(1) (x); _̣x f_2^(2) (x) ]
  
  is a constant function for x≠0.  We denote the constant by W[f_1,f_2].
  
  By a direct calculation, one can see that d/dxW[f_1,f_2]=0 for x≠0.
  Further, since
  
      [  1  0  0  0;  0  1  0  0; 2q  0  1  0;  0 2q  0  1 ]^t
      	[  0  0 -1  0;  0  0  0  1;  1  0  0  0;  0 -1  0  0 ][  1  0  0  0;  0  1  0  0; 2q  0  1  0;  0 2q  0  1 ]
      	=	[  0  0 -1  0;  0  0  0  1;  1  0  0  0;  0 -1  0  0 ],
  
  we have W[f_1,f_2](0+) = W[f_1,f_2](0-). 
  
  
  In the scalar equation case, one can investigate linear independence of two solutions by Wronskian. This is not the case for ODE systems (consider the vectors(1 0 0  0)^tand(0 1 0  0)^t).
  
   Fix ξ∈. For a pair of solutions f_1,f_2 ∈ X(ξ), consider the following two statements:
  
    * f_1 and f_2 are linearly dependent;
  
    * W[f_1,f_2]=0.
  
  Then, (i) implies (ii), but (ii) does not imply (i).
  
  
  We turn to the construction of the specific solutions to (<ref>).
  
  
  
   §.§ Construction of f_3 for x>0
  
  
  Let us first considerf_3, which should solve the integral equation
  
      f_3(x;ξ) = e^-μ x[ 0; 1 ]
      	- 2∫_x^ D_ξ (y-x) σ_3  𝒱(y) f_3(y;ξ)  dy
  
  forx>0.  Note thatf_1,f_2, andf_4will not solve the above integral equation; indeed, the the integral is not finite due to the asymptotic behavior of these solutions and the kernelD_ξ.
  
  We will solve the above equation forx∈. In addition to constructingf_3onx>0, we will use this solution in the construction off_1andf_2below. 
  
  
  For all ξ∈, there exists a unique solution f_3(x;ξ) to (<ref>) (on ) with the asymptotic condition
  
      e^μ x f_3(x;ξ) -
      	[ 0; 1 ]≲_ωμ^-1min(1,e^-γ x) .
  
  The solution is smooth in both variables and
  satisfies 
  
      |_̣ξ^l _̣x^k (e^μ x f_3(x;ξ))| ≲_ωμ^-l-1min(1, e^- 2γ x),
  
  for l,k ≥ 0 with l+k≥1.  Furthermore, the implicit constants depend continuously on ω. 
  
  The restriction of f_3(x;ξ) to the region x>0 coincides with any solution to (<ref>)
  which satisfies the above asymptotic condition for x>0.
  
  
  The proof of Proposition <ref> follows as in <cit.>.  We briefly recall the ideas:  To solve (<ref>), we introduce a modified kernel
  
      K_ξ (z) = e^-μ z D_ξ (z),
  
  which is bounded forz≥ 0.
  Then, (<ref>) is written as
  
      e^μ x f_3(x;ξ) =
      	[ 0; 1 ]
      	- 2∫_x^ K_ξ (y-x) σ_3 𝒱(y) e^μ yf_3(y;ξ)  dy.
  
  This is a Volterra equation, and the result follows from standard arguments. 
  
  
  
   §.§ Construction of f_1 and f_2 for x>0
  
  
  Let us next consider the solutionsf_1(x;ξ)andf_2(x;ξ), with our main focus onf_1(x;ξ). It is difficult to formulate (<ref>) as a Volterra type integral equation (like we did for (<ref>)), because the integral kernelD_ξgrows exponentially ifμ > γ.  Nonetheless, we can show the following.
  
  
  For all ξ∈, there exists a unique solution f_1(x;ξ) to (<ref>) with the asymptotic condition
  
      e^-i ξ x  f_1(x;ξ) -
      	[ 1; 0 ]≲_ωμ^-1x e^-x min (γ ,μ)
  
  for x> 0.
  Moreover, we have the estimate
  
      _̣ξ^l _̣x^k( e^-i ξ x  f_1(x;ξ)) ≲_k,l,ωμ^-1x^l+1 e^- x min (γ ,μ)
  
  for x >0 and for k,l≥ 0 with k+l≥ 1.  Furthermore, the implicit constants in the above inequalities depend continuously on ω.
  
  
  We again follow the strategy set out in <cit.> (see Lemma 5.3 therein).  We introduce the following ansatz:
  
      f_1(x;ξ) = [ f_1^(1)(x;ξ); f_1^(2)(x;ξ) ] := [ 1; 0 ] e^iξ x v(x;ξ) + [ f_3^(1)(x;ξ); f_3^(2)(x;ξ) ] u (x;ξ),
  
  whereuandvare new scalar unknowns.  Heref_3(x;ξ)is the solution to (<ref>) onconstructed in the previous section.
  To have the desired asymptotic behavior forf_1, we must impose that
  
      v(x;ξ) → 1,    f_3^(2)(x;ξ) u (x;ξ) → 0
  
  asx→.  We will eventually obtain the following estimate onv:
  
      |_̣ξ^l _̣x^k v(x;ξ) | ≲_k,lμ^-1x^l e^-γ x,
  
  which provides a slight improvement over the analogous estimate appearing in <cit.> (as we do not lose powers ofμunder differentiation with respect tox).
  
  We derive a system of equations foruandv.
  Forx>0, (<ref>) is the same as
  
      [ -1/2_̣x^2 -ξ^2/2 + V_1                    V_2;                   -V_2  1/2_̣x^2 - μ^2/2 -V_1 ]
      	f_1(x;ξ) = 0.
  
  As in Proposition <ref>, we consider the equation forx∈.
  
  We first show thatumay be obtained in terms ofv.  In view of (<ref>), the second coordinate of
   (<ref>) implies that
  
  
      u'(x;ξ) = -2 (f_3^(2)(x;ξ))^-2∫_x^ e^iyξ V_2(y)f_3^(2)(y;ξ)  v(y;ξ)  dy
  
  providedf_3^(2)(x;ξ)≠0.  Thus we have the following formula:
  
      u(x;ξ) = ∫_x_0^x 2 (f_3^(2)(y;ξ))^-2∫_y^ e^izξ V_2(z)f_3^(2)(z;ξ)  v(z;ξ)  dz  dy ,
  
  wherex_0 = x_0(ξ)will be chosen below. Note that this formula requiresf_3^(2)(x;ξ) ≠ 0for allx≥ x_0.
  
  We deviate from the approach of <cit.> in our choice ofx_0.  In <cit.>, the authors rely on the follow estimate (due to Proposition <ref>):
  
      e^μ x f_3(x;ξ) -
      	[ 0; 1 ]≲μ^-1max(1,e^-γ x).
  
  In particular, this estimate shows that there exists a constantx̃>0such that
  
      inf_ξ∈inf_x≥x̃ |e^μ x f_3^(2)(x;ξ)| ≥12.
  
  The authors of <cit.> then choosex_0(ξ)≡x̃. In our case, we rely on the fact that there existsR>0such that 
  
      inf_|ξ| ≥ Rinf_x∈ |e^μ x f_3^(2)(x;ξ)| ≥12,
   
  which implies that the choice ofx_0(ξ)for|ξ|≥ Rmay be taken arbitrarily. In fact, the authors of <cit.> already observed this and choosex_0=0for largeξ.  We take this one step further and choose negativex_0for largeξ.  This will allow us to derive the slightly improved estimate. 
  
  In summary, we choosex_0=x_0(ξ) ∈ C^()to be an even function such that
  
      x_0 = x_0(ξ)= x̃    |ξ| ≤ R , 
      
      	-1     |ξ | ≥ 2R
  
  and-1 ≤ x_0(ξ) ≤x̃for all other values ofξ, wherex̃>0andR>0are as above.
  
  We now need to derive an integral equation for the unknown scalarv.  From the first component of (<ref>), we obtain the relation
  
      - 1/2 v” - iξ  v' + V_1  v + e^-ixξ1/2 u”  f_3^(1)  + u' (f_3^(1))'  = 0.
  
  We may rewrite this equation as 
  
      v(x) = 1 + ∫_x^ E_ξ(y-x) [ 2V_1(y)v(y;ξ) + e^-iyξu”  f_3^(1)  + 2 u' (f_3^(1))'(y)  ] dy,
  
  where
  
      E_ξ(z) := ∫_0^z e^2isξ ds = e^2izξ-1/2iξ.
  
  Using (<ref>) we derive
  
      v(x) 
      	=    1 + ∫_x^  (K_1(x, y)+ K_2(x,y))  v(y;ξ)  dy,
  
  where 
  
      A(x;ξ) := 4/(f_3^(2)(x;ξ))^3(f_3^(2)(f_3^(1))'- (f_3^(2))'f_3^(1))(x;ξ),
  
      K_1(x,y) = 2 E_ξ(y-x) V_1(y) -f_3^(1)(y;ξ)/f_3^(2)(y;ξ) V_2(y),
  
  and
  
      K_2(x,y) = V_2(y)f_3^(2)(y;ξ) ∫_x^y E_ξ(z-x) A(z)  e^i(y-z)ξ dz .
  
  This is a Volterra-type equation. To complete the proof of Proposition <ref>, we need to constructvas a solution to (<ref>).
  
  To this end, we first collect some estimates onA.  The key fact is thatAhas the decaying factore^-μ x.
  
  
  For the solution f_3(x;ξ) to (<ref>) on  given in Proposition <ref>, we define
  A(x;ξ) as in (<ref>).
  Let x_0 be a number such that |f_3(x;ξ)|>0 holds for x>x_0.
  Let c:= inf_y≥ x_0|e^μ y f_3(y;ξ)|.
  Then, 
  
      |_̣ξ^l _̣x^k(e^-μ x A(x;ξ))| ≲_k,l,cμ^-l-1min (1, e^-2γ x)
  
  for k,l≥ 0 and x≥ x_0.
  If f_3(x;ξ)>0 for all x∈, the estimate is true for all x∈.
  
  We have the formula
  
      e^-μ x A(x) = 4/(e^μ x f_3^(2)(x;ξ) )^3 ((e^μ x f_3^(2) )(e^μ x f_3^(1))'-(e^μ x f_3^(2) )'(e^μ x f_3^(1) ))(x;ξ).
  
  Thus the lemma is a consequence of Proposition <ref>.
  
  
  We turn to problem of solving (<ref>).  We begin by estimating the integral kernels.  For now, let us restrict to the interval[x_0,∞). 
  
  We first note that
  
      |E_ξ (z)| ≤{   1/|ξ|,        |ξ| ≥ 1, 
          z,        |ξ| ≤ 1
      	.
  
   forz≥ 0.  Thus by Proposition <ref>, we have the following estimate forK_1:
  
      ∫_x^ |K_1(x,y)|  dy ≲∫_x^ |E_ξ (x-y)|e^-γ y dx ≲μ^-1 e^-γ x.
  
  
  Let us next considerK_2(x,y).  By Lemma <ref>,
  
      |A(x;ξ)| ≲μ^-1 e^(μ-2γ) x,
  
  and so
  
      ∫_x^ |K_2(x,y)| dy ≲∫_x^ e^-(μ+γ)y∫_x^y μ^-2 z e^(μ-2γ) z dz  dy
      	
      	≲μ^-2 e^-γ x
  
  forx≥ x_0.
  
  As (<ref>) is a Volterra-type equation, we may now obtain a solutionv(x;ξ)on[x_0,)by standard arguments (see e.g. <cit.>). Furthermore, we can obtain
  
      |v(x;ξ) -1 | ≲∫_x^ (|K_1(x,y)|+ |K_2(x,y)|)  dy  ≲μ^-1 e^-γ x
  
  on this interval.
  
  We next estimatex-derivatives, still restricting to the interval[x_0,∞).  As we claim a small improvement over <cit.>, we will estimate in some detail. Letk≥1andl=0and observe that 
  
      _̣x^k∫_x^  K_1(x, y) v(y;ξ)  dy 
      
      	   = ∑_j=0^k-1∫_x^ K_1^(k,j)(x,y) _̣y^j v(y,ξ)  dy + ∫_x^ K_1(x,y) _̣y^k v(y,ξ)  dy,
  
  where
  
      K_1^(k,j)(x,y):= 2
      
      	[ k; j ]
      	 E_ξ(y-x) _̣y^k-jV_1(y) -f_3^(1)(y;ξ)/f_3^(2)(y;ξ) V_2(y)
  
  for0≤ j ≤ k-1.
  
  By Proposition <ref>, we have
  
      ∫_x^ |K_1^(k,j)(x,y)| ≲_k ∫_x^ E_ξ(y-x) e^-γ y dy ≲μ^-1 e^-γ x
  
  for0≤ j ≤ k-1.
  We also remark that
  
      _̣x^k  ∫_x^  K_2(x, y) v(y;ξ)  dy 
         = ∑_j=0^k∑_m=0^min (j,k-1)∫_x^ K_2^(k,j,m)(x,y) _̣y^m v(y;ξ)  dy + ∫_x^ K_2(x,y)_̣y^k v(y;ξ) dy,
  
  where
  
      K_2^(k,j,m)(x,y) := 
      
      	[ k; j ][ j; m ]_̣y^j-m (V_2(y)e^μ yf_3^(2)(y;ξ))
      ×∫_x^y E_ξ(z-x) (_̣z^k-j(e^-μ z A(z)))  e^(y-z)(iξ-μ) dz .
  
  By Proposition <ref> and by assumption, we have the estimate
  
      |_̣x^k (V_2(x)e^μ xf_3^(2)(x;ξ))| ≲_k e^-γ x.
  
  Recalling Lemma <ref>, we arrive at the estimate
  
      |K_2^(k,j,m)(x,y)| ≲_k e^-γ y∫_x^y μ^-1 (z-x) μ^-1 e^-2γ z e^-μ(y-z) dz
      	≲μ^-2 e^-γ y.
  
  Combining the above identities, one sees that_̣x^k v(x;ξ)solves a Volterra-type equation, and hence we obtain the following estimate by standard arguments: 
  
      |_̣x^k v(x;ξ) | ≲_k μ^-1 e^-γ x
  
  forx≥ x_0.  The casel ≥ 1is handled similarly.
  
  We next estimateu(given byvas above) and complete the proof of Proposition <ref>.  We rely on the following lemma, which follows from direct computation.
  
  
  Let μ,γ>0 and x_0∈. For n,m∈ℕ∪{0}, we have 
  
      ∫_x^ (z-x)^n e^-μ(z-x) e^-γ z dz
      	≲_n  μ^-n-1  e^-γ x
  
  for x∈, and
  
      ∫_x_0^x    (x-y)^n e^μ(y-x)∫_y^ (z-y)^m e^-μ (z-y) e^-γ z dz  dy
         ≲_m,n,γ,x_0μ^-(m+n)-2x^n+1 e^-x min(μ,γ)
  
  for x≥max(0,x_0).
  
  
  With all of the estimates above in place, we can turn to the proof of Proposition <ref>. 
  
  
  We define x_0 as in (<ref>).
  Then, one obtains u by the formula (<ref>).  Furthermore, by the estimates on v and by Lemma <ref>,
  
      |f_3(x;ξ) u(x;ξ)| 
      	   ≲∫_x_0^x e^μ (y-x)∫_y^ e^-μ ( z-y) e^- γ z dz  dy
         ≲μ^-2x e^-x min (γ,μ) .
  
  
  Thus, the pair (u,v) gives us the solution f_1(x;ξ) which satisfies
  
      f_1(x;ξ) - [ e^ixξ;     0 ]≲μ^-1x e^-min(γ,μ) x
  
  for x>x_0(ξ). 
  For ξ such that x_0(ξ)>0, we extend the solution to the region 0< x< x_0(ξ) by the general existence theorem.
  
  We turn to the estimate of derivatives of the solution.  In particular, we need to estimate derivatives of u.
  
  We introduce
  
      ũ(x)
      	   :=e^-(iξ + μ)x u(x) 
         = 
      	∫_x_0^x 2 (e^μ y f_3^(2)(y;ξ))^-2 e^(y-x)(iξ +μ)
                 ×∫_y^ e^(z-y)(iξ-μ) V_2(z)(e^μ zf_3^(2)(z;ξ))  v(z;ξ)  dz  dy
  
  We may also write
  
      ũ(x)
      	   = 
      	∫^x-x_0_0  e^-w(iξ +μ) H(x-w)  dw ,
  
  where
  
      H(z) := 2 (e^μ z f_3^(2)(z;ξ))^-2∫_0^ e^a(iξ-μ) V_2(a+z)(e^μ (a+z)f_3^(2)(a+z;ξ))  v(a+z;ξ)  da .
  
  This expression yields
  
      _̣x^kũ(x)
      	   = ∑_j=0^k-1 H^(j)(x_0) e^x_0(iξ + μ) (-(iξ+μ))^k-1-j e^-x(iξ + μ)
            + ∫_0^x-x_0 H^(k)(x-w) e^-w(iξ +μ) dw
         =: I_1 + I_2
  
  for k≥1. 
  
  We now observe that the derivative of the first term is bounded by
  
      |_̣ξ^l I_1(x,ξ)| ≤ C_k,lx^l μ^k-1 e^x_0 μ e^-xμ≲_k,R e^-1/2μx^l e^-xμ
  
  for |ξ|≥ 2R and x>0, where we have used x_0=-1. For |ξ| ≤ 2R and x ≥ x_0(ξ), we use boundedness |ξ| ≤ 2R and x_0 ≤x̃ to obtain the same conclusion.
  
  Let us next consider derivatives of I_2.  If any derivative lands on e^μ x f_3^(2)(x;ξ), then we gain μ^-1, while if it lands on v(x;ξ) or e^x(iξ±μ) then we lose ⟨ x⟩.  Hence, 
  
      |_̣ξ^l I_2(x)| ≲   ∑_j=0^l ∫_0^x-x_0 w^j e^-μ w∫_0^ (a+w)^l-j e^-μ a e^-γ (a+w) da  dw 
      
      	=   ∑_j=0^l ∫_x_0^x (x-y)^j e^μ(y-x)∫_y^ (z-y)^l-j e^-μ (z-y) e^-γ z dz  dy.
  
  Using Lemma <ref>, we find
  
      |_̣ξ^l  I_2(x)| ≲μ^-l-1x^l+1 e^-x min (γ, μ)
  
  for x≥ 0. This yields the desired estimate for |ξ| ≥ 2R or x ≥ 0.  Note that the above argument also works  for k=0 and l≥1.
  
  Finally, we consider the case |ξ| ≤ 2R, x_0(ξ) > 0, and x ∈ [0,x_0(ξ)].
  Notice that this case both x and ξ is bounded.
  Hence, it is sufficient to use the following estimate for a general solution:
  
      |_̣ξ^l _̣x^k f_1(x;ξ)| ≲_k,l,R 1.
  
  This completes the proof.
  
  
   §.§ Construction of f_4 for x>0
  
  
  Let us next construct the solutionf_4(x;ξ).  We first need to deal with the fact that the desired asymptotic behavior, namely
  
      f_4(x;ξ) → e^μ x[ 0; 1 ]asx→∞,
  
  does not uniquely specify the solution.  Indeed, if one finds one such solution then any linear combination of the solution and the solutionsf_1,f_2, andf_3satisfies the same asymptotics.
  
  Following <cit.>, we will look forf_4obeying the Wronskian condition
  
      W[f_1,f_4] = W[f_2,f_4] = 0.
  
  As we already have
  
      W[f_1,f_3] = W[f_2,f_3] = 0,
  
  the above condition determines the `coefficients' off_1andf_2.  We remark, however, that we still have the freedom to add a scalar multiple off_3.
  
  To obtain a solution, we solve the following integral equation:
  
      f_4(x;ξ) =    e^μ x[ 0; 1 ] + ∫_x^[             0             0;             0 -1/μe^μ (x-y) ]𝒱(y) f_4(y;ξ)  dy 
         +	∫_x_1^x 
      	[   2sinξ(x-y)/ξ              0;              0 -1/μe^-μ (x-y) ]𝒱(y) f_4(y;ξ)  dy
  
  with a suitable choice ofx_1=x_1(ξ) ∈.
  We denote the particular solution we construct byf̃_4(x;ξ).  Similar to the construction off_1andf_2, we will choose negativex_1for largeξ.  More precisely, we definex_1=x_1(ξ) ∈ C^(), with constantsx̃≥ 0 andR=R(𝒱_L^1(),ω,γ)> 0to be chosen later, to be an even function satisfying
  
      x_1(ξ) = 
      	x̃    |ξ| ≤ R
      
      	-1     |ξ| ≥ 2R
  
  and-1 ≤ x_1(ξ) ≤x̃for anyξ∈.
  
  
  There exists a bounded function x_1 = x_1(ξ) such that (<ref>) admits a unique solution f̃_4(x;ξ)
  obeying 
  
      f̃_4(x;ξ) ≤ 2 e^μ x
  
  for x≥ x_1 and
  
      e^-μ x f̃_4^(j)(x;ξ) - δ_2j≲_ωμ^-(3-j)x^2 e^-x min (γ ,μ)
  
  for j=1,2, x ≥ x_1, and ξ∈.  Here δ_ij denotes the Kronecker delta.  Moreover, we have the estimate
  
      _̣x^k (e^-μ x f̃_4^(j)(x;ξ)) ≲_k,ωμ^-(3-j)x^2 e^- x min (γ ,μ)
  
  for x ≥ x_1 and for k≥ 1.  The implicit constants in the above inequalities depend continuously on ω.
  
  
  As before, our choice ofx_1affords us some slight improvement compared to the estimates appearing in <cit.>.  In particular, the power ofμdoes not grow withk, and the estimate for the first component is improved.
  
   Writing g(x;ξ) := e^-μ x f_4(x;ξ), we arrive at the equation
  
      g(x;ξ) =   [ 0; 1 ] + ∫_x^[     0     0;     0 - 1/μ ]𝒱(y) g(y;ξ)  dy 
         +	∫_x_1^x 
      	[ 2sinξ(x-y)/ξe^-μ (x-y)                      0;                      0       - 1/μe^-2μ (x-y) ]𝒱(y) g(y;ξ)  dy .
  
  Denoting the right hand side by Ψ[g](x), we will show that Ψ is a contraction map on the complete metric space (X,d) defined by
  
      X    := { g ∈ C([x_1,); ^2 ) : g_L^([x_1,))≤ 2 },
      
      	d(g_1,g_2)    := g_1-g_2_L^([x_1,)).
  
  Note that the integrals have bounded kernels since
  
      sinξ(x-y)/ξe^-μ (x-y)≤ (x-y)e^-μ (x-y)≤μ^-1sup_w ≥ 0 w e^-w
  
  for 0 ≤ y ≤ x.
  
  Now let g,g_1,g_2 ∈ X. For x∈ (x_1,), we have
  
      |Ψ[g](x)| ≤    1 + ∫_x^μ^-1 C |𝒱(y)|  dy + ∫_x_1^x μ^-1 C |𝒱(y)|  dy
      ≤    1 + Cμ^-1𝒱_L^1([x_1,))
  
  and similarly
  
      |Ψ[g_1](x)-Ψ[g_2](x)| ≤ Cμ^-1𝒱_L^1([x_1,))g_1 - g_2_L^ ([x_1,)).
  
  
  We now observe that there exists x̃≥ 0 such that Ψ is a contraction for all ξ∈ with the choice  x_1=x̃.  Moreover, there exists R=R(𝒱_L^1())≥ 0 such that if |ξ| ≥ R then Ψ is a contraction for any x_1 ∈. Without loss of generality, we may suppose that R ≥max(ω,γ) so that |ξ| ∼_ωμ≥ 2γ for |ξ|≥ 2R.  Now, we define x_1(ξ) by (<ref>) with this choice of x̃ and R.  We then obtain a unique solution g ∈ X to the equation g=Ψ[g].
  
  We next estimate derivatives of g. We will show
  
      _̣x^k (e^-μ x f̃_4(x;ξ)) ≲_k,γμ^-1x^2 e^- x min (γ ,μ)
  
  for x ≥ x_1 and for k≥ 1.  We first have
  
      g^(k)(x;ξ)
          = ∫_0^x-x_1[ 2cos (ξ w) e^-μ w-2μsinξ w/ξe^-μ w                                  0;                                  0                          2 e^-2μ w ] (_̣w^k-1(𝒱 g))(x-w) dw 
           + ∑_j=0^k-2_̣x^k-1-j[ 2sinξ (x-x_1)/ξe^-μ (x-x_1)                           0;                           0           -1/μe^-2μ (x-x_1) ] (_̣w^j(𝒱 g))(x_1)
  
  for k≥1, with the convention that the second term is zero when k=1.
  
  Consider the case |ξ|≥ 2R.   We will show by induction that
  
      |g^(k)(x)| ≲_k μ^-1 e^-γ x
  
  holds for k≥1 and x≥ x_1=-1. The base case is immediate from (<ref>). Indeed, since μ≥ 2γ holds in this case, we have
  
      |g'(x)| ≲ ( ( 1_{w≥ 0} e^-μ w)* ( 1_{w≥ -1} e^-γ w))(x)
      	≤ 2μ^-1 e^-γ x
  
  for x≥ -1.  Now suppose that the above bound is true for 1 ≤ k ≤ k_0.  Then we derive
  
      _̣w^k_0 (𝒱 g(·;ξ))(z) dw≲_k_0,γ  e^-γ z
  
  for z ≥ 0.
  
  Hence, as in the base case, we have
  
      |g^(k_0+1)(x)| ≲ ( ( 1_{w≥ 0} e^-μ w)* ( 1_{w≥ -1} e^-γ w))(x) + e^-1/2μe^-μ x≲μ^-1 e^-γ x
  
  from (<ref>). Here the choice x_1=-1 is used for the estimate of the second term.
  
  For |ξ| ≤ 2R and x ≥ x_1, we have
  
      |g^(k)(x)| ≲_k x^2min(1, e^-x min(μ,γ) ) ≲_γ 1
  
  by a similar induction argument.  The difference is that we need to use the worse estimate
  
      [ cos (ξ w)e^-μ w - μsinξ w/ξe^-μ w                                 0;                                 0                        -2 e^-2μ w ]≲_R w e^-μ w
  
  for w≥ 0. The base case k=1 follows from
  
      |g'(x)| ≲ ( ( 1_{w≥ 0}w e^-μ w)* ( 1_{w≥ -1} e^-γ w))(x)
      	≲x^2  e^-x min (γ,μ) ≲_γ 1.
  
  If the estimate is true up to k_0, we have
  
      _̣w^k_0 (𝒱 g(·;ξ))(z) dw≲_k_0,γ  e^-γ z
  
  for z≥ 0.
  Hence,
  
      |g^(k_0+1)(x)|    ≲ ( ( 1_{w≥ 0}w e^-μ w)* ( 1_{w≥ -1} e^-γ w))(x) + e^-1/2μx e^-μ x
         ≲x^2 e^- x min(γ,μ)
  
  as desired.
  
  
  Finally, we consider an improved estimate for the first component. We first observe that g satisfies the ODE
  
      g” = -2μ g' -(ξ^2 +2E)(Id+σ_3) g + 2 σ_ 3𝒱 g.
  
  Thus
  
      |g_1^(k)| ≤μ^-2|g^(k+2)| + 2μ^-1 |g^(k+1)| + 2 μ^-2 |_̣x^k(𝒱 g)|
  
  for k≥ 0, where g_1 is the first component of g. 
  Combining the above estimates, we complete the proof of Proposition <ref>.
  
  We have similar estimates for ξ-derivatives. 
  We omit the details and only remark that the equation for _̣ξ^l g is of the same form as that for g.
  As a result, we obtain 
  
      |_̣ξ^l (e^-μ xf̃_4(x;ξ))| ≲μ^-2x^l e^-x min (γ,μ)
  
  for l≥ 1, for instance. One also has estimates on _̣x^k _̣ξ^l (e^-μ xf̃_4(x;ξ)) (k,l≥1)
  by proceeding as in the case l=0.
  
  We conclude this section by demonstrating thatg-1decays asx→.  Asgis a bounded solution, the equation forgyields
  
      |g(x)-1| ≲   ∫_x^μ^-1 |𝒱(y)|  dy + ∫_x_1^x [ 2sinξ(x-y)/ξe^-μ (x-y)                      0;                      0        -1/μe^-2μ (x-y) ]|𝒱(y)|  dy.
  
  It is straightforward to see that the first term isO(μ^-1 e^-γ x)for largex.
  We estimate the second term as follows: 
  
      ∫_x_1^x (x-y) e^-μ (x-y) e^-γ y dy
      	= e^- γ x∫_x_1^x (x-y) e^-(μ-γ) (x-y) dy,
  
  which is bounded by
  
      e^- γ x∫_0^ w e^-(μ/2) w dw = C_γμ^-2 e^-γ x
  
  ifμ≥ 2γ, and by
  
      e^-γ xmax (1, e^-(μ-γ) (x-x_1)  ) ∫_x_1^x (x-y)   dy ≲_γx^2 e^-x min (γ, μ)
  
  ifμ≤ 2γ.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Construction of solutions for x<0
  
  
  So far, we have constructed solutionsf_j(x;ξ)(j=1,2,3,4) to (<ref>) forx>0.
  We now consider the continuation tox<0.  Note that we already have four linearly independent solutionsf_j(-x;ξ)(j=1,2,3,4) whenξ≠0, and so the continued solution is given by a linear combination of these forx<0.  However, it it is not clear whether we have a uniform-in-ξbound on the coefficients, asf_1andf_2coincide whenξ=0.
  
  In order to get a uniform-in-ξbound, we solve the integral equation
  
  
  
      f_j(x;ξ) =    D_ξ' (x) f_j(0;ξ) + D_ξ (x)  f_j'(0-;ξ) 
         -2 ∫_x^0 D_ξ(y-x) σ_3 𝒱(y)  f_j(y;ξ)   dy.
  
  forx<0.  We summarize the result as follows:
  
  
  For all ξ∈, the solution f_j(x;ξ) (j=1,2,3,4) to (<ref>) given above
  can be extended as solutions on the whole line x∈. 
  The solution is smooth in (x,ξ) ∈ (∖{0}) × and
  satisfies 
  
      | _̣ξ^l _̣x^k f_j(x;ξ)| ≲μ^-1+kx^l e^μ |x|
  
  for x<0 and k,l ≥ 0. 
    One sees that e^-μ|x| f_j(x) solves a Volterra type integral equation on x∈(-,0).
  This can be solved by standard arguments.  For the data, we have
  
      _̣x^k f_j (0-;ξ) =  O(μ^-1+k)
  
  for k≥ 0 and j=1,2,3,4.
  Indeed,
  thanks to estimates for solutions on x>0, we know
  
      _̣x^k f_j (0+;ξ) =  O(μ^-1+k)
  
  for k≥ 0 and j=1,2,3,4. Then, by the jump condition we obtain the desired bound for k=0,1.
  The differential form of the equation gives us the bound for k≥2.
  
  
  
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