Straight quantum layer with impurities inducing resonances
Sylwia Kondej





Straight quantum layer  with impurities inducing resonances  
     Sylwia Kondej
    
=============================================================





Institute of Physics, University of Zielona Góra, ul.
Szafrana 4a,  65246 Zielona Góra, Poland





e-mail: skondej@if.uz.zgora.pl,


We consider a straight three dimensional quantum layer
with singular potential supported on a straight wire which is
localized perpendicularly to the walls and  connects them. We prove
that the infinite number of  embedded eigenvalues appears
in this system. Furthermore, we show that after introducing
a small surface impurity to the layer, the embedded eigenvalues turn to
the second sheet resolvent poles which state
resonances. We discuss the asymptotics of the imaginary component of the resolvent
pole with respect to the surface area.








Keywords: Singular perturbations,
embedded eigenvalues, resonances.



§ INTRODUCTION


The paper belongs to the line of  research often called Schrödinger operators with delta potentials
[In the following we will equivalently use the notations delta interaction and delta
potential.].
The analysis of these type of potentials is motivated by mesoscopic physics systems with the semiconductor
structures  designed in such a way that they  can be mathematically modelled by the Dirac delta  supported on the sets
of lower dimensions. The support of delta potential imitates the  geometry of the semicondutor material, for example,
it can take a form of one dimensional sets (wires) or surfaces with specific geometrical properties. A particle is confined
in the semiconductor  structure however the model admits a possibility of tunneling.
Therefore these types of systems are called in literature leaky quantum graphs or wires. One of the most
appealing problem in this area is the question  how the geometry of a wire affects the spectrum;
cf. <cit.> and <cit.>. The aim of the present paper is to discuss
how the surface perturbation leads to resonances.

We consider a non relativistic three dimensional  model of  quantum  particle
confined between two infinite unpenetrable parallel walls which  form
a straight quantum layer defined by Ω :={(x, x_3)∈^2 × [0, π ]}.
 In the case of absence
of any additional potential  the Hamiltonian of such system is given  by the negative Laplacian
 -Δ : D(Δ )
 : → L^2 (Ω ) with the domain D(Δ )=W^2,2(Ω )∩ W^1,2_0(Ω ), i.e. with  the
Dirichlet boundary conditions on ∂Ω.
The spectrum of   -Δ is determined by σ_ess(-Δ )=[1, ∞ ) however it is useful to keep
in mind that the energies in x_3-direction  are quantized and given by {k^2}_k=1^∞.

At the first stage we introduce a straight wire I which connects  the walls ∂Ω
being at the same time perpendicular to them. We assume
the presence of interaction localized on I and characterized by the coupling constant α∈. The symbolic Hamiltonian of
such system can be formally written

    -Δ + δ_α , I ,

where δ_α ,I represents delta potential supported on I.  Since the interaction support in this model has the co-dimension
larger then one it is called strongly singular potential.
 The proper mathematical definition of Hamiltonian
can be  formulated  in the terms of boundary conditions. More precisely, we define H_α as  a self adjoint extension
of -Δ|_C^∞_0 (Ω∖ I). determined by means of the appropriate boundary conditions which functions
from the domain D(H_α )
 satisfy on I.
 The coupling constant α is involved in the mentioned boundary
conditions, however, it is worth to say at this point that
 α does not contribute  additively to the structure of Hamiltonian.

 To describe spectral  properties of H_α we can rely on radial symmetry of the system
and consider two dimensional system with point interaction governed by the Hamiltonian H^(1)_α.
The spectrum of H^(1)_α consists of positive half line and one discrete negative eigenvalue

    ξ_α = -4^2 (-2πα +ψ(1)) ,

cf. <cit.>, where -ψ(1)= 0,577... determines the Euler–Mascheroni constant.
This reflexes the structure of spectrum of H_α, namely,
for each l∈ the number

    ϵ_l = ξ_α +l^2 ,

gives rise to an eigenvalue of H_α. Note that infinite number  of ϵ_l lives above the threshold of the
essential spectrum and, consequently, the Hamiltonian H_α admits  the infinite number of embedded eigenvalues.
 
 In the second stage we introduce to the layer
 an attractive interaction supported on a finite C^2 surface Σ⊂Ω separated
 from a wire I by some distance, cf. Fig. 1.  Suppose that  β≠ 0 is a real number. The Hamiltonian H_α , β which governs this system can
 be symbolically written  as
 
    -Δ +δ_α, I -βδ_Σ ,

 where δ_Σ stands for the  Dirac delta supported on Σ; this term represents weakly
 singular potential
  Again, a proper mathematical definition of
 H_α, β can be formulated as a self adjoint extension of 
    H_α|_{f∈D(H_α )
    ∩ C^∞ (Ω∖ I)  :  f =0    on   Σ}.  .


 
 This extension is defined by means of the
   appropriate boundary conditions on Σ discussed in Section <ref>.




 The aim of this paper is to analyse how the presence of surface interaction supported on
 Σ affects the embedded eigenvalues. The existence of embedded eigenvalues is a direct consequence
 of the symmetry. By introducing additional interaction on Σ we break this symmetry, however, if the perturbation
 is small then we may expect that the system preserves a ”spectral memory“ on original eigenvalues.
 In Section <ref> we show, for example,  that
  if the area |Σ| → 0  then the embedded eigenvalues ϵ_l
 turn to complex poles of the resolvent
  of H_α, β. These poles are given by  z_l = ϵ_l +o(|Σ | ) with  z_l <0; the latter confirms that z_l is localized on the second sheet continuation.
  We derive the explicit formula for the lowest order of the imaginary component of z_l and show that it admits  the  following asymptotics
  
    z_l = 𝒪 (|Σ|^2) .
 The poles of resolvent state resonances in the system governed by H_α , β
  and z_l is related to the width of the resonance given by -2 z_l.
  
 

Finally, let us mention that various  types of resonators in waveguides and layers  have been already analyzed.
For example, in <cit.> the authors study resonances induced
by the twisting of waveguide  which is responsible for breaking symmetry.
The planar waveguide with narrows playing the role of resonators
has been studied in <cit.>. On the other hand, the straight Dirichlet or Neuman waveguides with windows or barriers
inducing resonances have been analyzed in <cit.>.
Furthermore, resonances in the curved  waveguides with finite branches
have been  described in <cit.>. It is also worth to mention that quantum waveguides with electric and magnetic fields
have been   considered,  cf. <cit.>.

On the other hand, the various types of resonators induced by delta potential
in two or three dimensional systems
have been analyzed. Let us mention the results of <cit.>
which describe resonances  in the terms of breaking symmetry parameters or by means of tunnelling effect.

In  <cit.> the authors consider a straight two dimensional waveguide with a semitransparent perpendicular
barrier modeled by delta  potential. It was shown that after changing slightly the slope of barrier the embedded eigenvalues turn to resonances;
the widths of these  resonances can be expressed in the terms of the barrier slope.
The present paper is, in a sense, an extension of <cit.>. However,  the strongly singular character of the delta
interaction supported on I causes that even an infinitesimal  change of the slope of I, can not be understood as a small perturbation.
Therefore  the resolvent poles are not interesting from the physical point of view
since they rapidly escape far away
from the real line.
In the present model the role of
small perturbation plays  delta potential on Σ which
 leads to resonances.


Finally, it is worth to mention that the spectral properties of quantum waveguides and layers with delta
interaction have been studied, for example, in  <cit.>. The results of <cit.>
concern weakly singular potentials and in <cit.> the authors consider  strongly singular
interaction. In the present paper we combine both  types of delta interaction and analyze how they affect
each other.









  

General notations:

 ∙  stands for the complex plane and _± for the upper, respectively,  lower half-plane.



∙ ·, (· ,· ) denote the norm and the scalar product
in L^2 (Ω ) and (·, · )_Σ defines the scalar product in  L^2 (Σ ).

∙ Suppose that A stand
for a self adjoint operator. We standardly denote by σ _ess(A),
 σ _p(A) and ρ (A), respectively, the essential spectrum, the point spectrum and the resolvent set of A.



∙ The notation C stands for a constant which value can vary from line
to line.



§ PARALLEL WALLS CONNECTED BY WIRE INDUCING  EMBEDDED EIGENVALUES





 §.§ Free particle in layer.

Let Ω⊂^3 stand for  a layer defined
 by Ω := { x= ( x, x_3) : x∈^2   ,x_3 ∈ [0, π ]} and in the following we assume convention
 x = (x_1, x_2)
 ∈^2. 

 The ”free“ Hamiltonian is determined by  
    H= -Δ : D(H)= W^ 2,2 (Ω )∩ W_0 ^1,2(Ω )→ L^2 (Ω )
 and it admits the following decomposition
 
 
   
    H=-Δ ^(2)⊗ I +I ⊗ -Δ^(1)   on    L^2 (^2) ⊗ L^2 (0,π ) ,

 where Δ^(2) : D(Δ^(2)) = W^2,2(^2)→ L^2  (^2) stands for the
 two-dimensional Laplacian and Δ^(1) : D(Δ^(1)) = W^2,2(0,π)∩ W_0^1,2(0,π)→
 L^2 (0,π) determines one-dimensional Laplacian with the Dirichlet boundary conditions.

 To define the resolvent  of H  it is useful to note that
 the sequence {χ_n}_n=1^∞ given by   
    χ_n (x_3) : = √(2/π)sin ( n x_3) ,    n∈

 forms an
 orthonormal basis in L^2 (0, π). Suppose that z∈∖ [1,∞ ). Then
R(z):= (-Δ -z)^-1 defines an integral operator with the kernel
 
    𝒢(z; x,  x',x_3,x'_3):= 1/2π∑_n=1^∞ K_0 (κ_n (z)
       |x- x'|) χ_n (x_3)χ_n (x_3 ' ) ,

 where K_0 (·) denotes the Macdonald function, cf. <cit.>, and
 
    κ _n (z):= -i√(z-n^2) ,   √(z-n^2) >0  .

In the following we will also use the abbreviation 𝒢 (z) for (<ref>). The threshold of spectrum of H is determined by the lowest discrete transversal energy, i.e. 1.
 Moreover, it is purely absolutely continuous and consequently, it takes the form
 
    σ (H) = [1,∞ ) .

 

 §.§ Layer with perpendicular wire: embedded eigenvalues phenomena.

We introduce a  wire defined by the straight segment of width π and perpendicular to walls.
 The presence of the wire will be modelled by delta interaction supported on I⊂Ω, where
  I:= (0,0)× [0,π].

 

 In view of the radial symmetry the operator with delta interaction on I admits a natural
 decomposition on
 L^2 (Ω)= L^2 (^2)⊗ L^2 (0,π ) and acts in the subspace L^2 (^2) as the Schrödinger operator
 with one point interaction. Therefore, the  delta potential can be determined by appropriate boundary conditions, cf. <cit.>,
 which can be implemented, in each sector of the transversal
energy, separately. For this aim we decompose a function  ψ∈ L^2 (Ω ) onto
ψ (x)=∑_n=1^∞ψ_n (x )χ_n (x_3), where
 ψ_n (x ):= ∫_0^πψ (x, x_3 ) χ_n (x_3)dx_3.

 
   
  D_1)  We say that a function ψ belongs to the set D'⊂
 W^2,2_loc (Ω∖ I) ∩ L^2 (Ω ) if Δψ∈ L^2 (Ω ),  ψ|_∂Ω=0 and the following limits
 
    Ξ_n (ψ ):= - lim_|x|→ 0  1/ln |x|ψ_n (x) ,     Ω_n (ψ):=
     lim_|x|→ 0  ( ψ_n (|x |) - Ξ_n (ψ )ln |x |)

 are finite.
   
  D_2) For  α∈, we  define the set
 
    D (H_α ):= {ψ∈ D'  :  2παΞ_n (ψ ) = Ω_n (ψ)   for   any   n∈}

 and the operator H_α : D (H_α ) → L^2 (Ω ) which acts
 
    H_αψ (x)= -Δψ (x) ,  for    x∈Ω∖ I .

 
The resulting operator H_α : D(H_α )→ L^2 (Ω )  coincides
   
    -Δ_α  ^(2)⊗ I +I ⊗ -Δ^(1)   on    L^2 (^2) ⊗ L^2 (0,π ) ,

 where Δ_α ^(2) : D(Δ_α ^(2)) → L^2  (^2) stands for the
 two-dimensional Laplacian with point interaction, cf. <cit.> with the domain D(Δ_α ^(2)). Consequently, H_α

 
  is self adjoint and its spectral properties will be discussed in the next section.



 §.§ Resolvent of H_α.
 

Suppose that z∈_+.
We  use the standard notation R_α (z) for the resolvent operator, i.e. R_α (z):= (H_α - z)^-1.
To figure out the explicit resolvent formula we introduce

    ω_n (z; x):= 1/2π K_0 (κ_n (z)| x | ) χ_n (x_3) ,    n∈ ;

in the following we will use also abbreviation ω_n (z)= ω_n (z; · ).

The following theorem states the desired result.
 The essential spectrum of H_α is given by

    σ_ess (H_α )=[1, ∞ ) .

Furthermore, let [Analogously as in the previous discussion
 we assume √(z-n^2) >0. The logarithmic function z↦ln z is defined  in the cut
 plane -π < z <π
 and admits continuation to entire logarithmic Riemann surface.]

    Γ_n (z):= 1/2π( 2πα +s_n (z)) ,  where    s_n (z):=-ψ(1) +ln√(z-n^2)/2i .

Suppose that z∈∖ [1,∞ ) and Γ _n (z) ≠ 0. Then z∈ρ (H_α ) and operator R_α (z) admits the Krein-like form:

    R_α(z)= R(z) +∑_n=1^∞Γ_n (z)^-1  (ω_n (z̅ ), · )ω_n(z) .



Our first aim is to show that (<ref>) defines the resolvent of H_α.
Operator H_α is defined as the self adjoint extension of -Δ |_C^∞ _0 (Ω∖ I).
Suppose that f∈ C^∞ _0 (Ω∖ I). Then g:=(-Δ -z )f∈ C^∞ _0 (Ω∖ I).
Employing the fact that ω_n (z)=𝒢(z)∗ (δχ_n ),
where 𝒢(z) is the kernel defined
by (<ref>) and δ =δ (x),
we  conclude  that (ω_n (z̅ ),  g)= ⟨δχ_n, f⟩_-1,1 = 0
 where  ⟨· , ·⟩_-1,1 states the duality
between W^-1,2(Ω ) and W^1,2(Ω ). This, consequently,
 implies
R_α (z) (-Δ -z )f = R (z) (-Δ -z )f= f in view of
(<ref>) which means that R_α (z) defines the resolvent of a self adjoint
extension of -Δ |_C^∞ _0 (Ω∖ I). To complete the proof we have to show that
any function g=R_α (z)f satisfies boundary conditions (<ref>).
In fact, g admits the unique decomposition g=g_1+g_2, where g_1:= R(z)f and
g_2= ∑_n=1^∞Γ_n (z)^-1  (ω_n (z̅ ), f )ω_n(z).
Therefore, a nontrivial contribution to Ξ_n (g) comes from g_2 since g_1 ∈ W^2,2(Ω ).
Employing the asymptotic behaviour
of the Macdonald function, cf. <cit.>

    K_0 (ρ)=  ln1/ρ +ψ(1)+𝒪(ρ) ,

 we get Ξ_n (g)= 1/2πΓ_n  (z)^-1(ω_n (z̅), f) and
 
    Ω_n (g)= (1-  1/2πΓ_n  (z)^-1s_n (z) ) (ω_n (z̅), f) =
     αΓ_n  (z)^-1 (ω_n (z̅), f) .

 Using (<ref>) one obtains  (<ref>). This completes the proof of (<ref>).
 The stability of the essential spectrum can be concluded in the analogous way as in <cit.>. The key step
 is to show that R(z)-R_α (z) is compact. The statement can be proved relying on
compactness of the trace map S :  
W^2,2(Ω )→ L^2 (I ) which follows from the boundedness of the trace map, cf. <cit.> and the compactness theorem, cf. <cit.>.
 
 This implies, in view of boundedness of R(z) :  L^2 (Ω )→  W_0^1,2(Ω )∩ W^2,2(Ω ), that
 SR(z) :  L^2 (Ω )→ L^2(I) is compact. Employing the resolvent formula, cf. <cit.>, and the fact
 that the remaining operators contributing to
   R(z)-R_α (z)  are bounded we conclude that     R(z)-R_α (z)  is  compact.
 
 




 The spectral analysis developed in this work is mainly   based
on  the resolvent properties. In the following we will use the results of
<cit.> where strongly as well as weakly singular potentials were considered.



In the following theorem we state the existence of eigenvalues of H_α.
 
  Let 𝒜_α := {n∈ : ξ_α +n^2 <1 }. Each ϵ_n:= ξ_α +n^2, where n∈𝒜_α defines the discrete
  eigenvalue of H_α with the corresponding eigenfunction ω_n:= ω_n (ϵ_n). In particular,  this means that for any  α
  operator H_α has at least one eigenvalue ϵ_1 below the threshold of the essential spectrum.
  
 Operator H_α has infinite number of embedded eigenvalues. More precisely, for any n∈∖𝒜_α
  the number ϵ_n:= ξ_α +n^2 determines the embedded eigenvalue. In particular,
  there exists ñ∈∖𝒜_α
  such that ϵ_n ∈ ((n-1)^2, n^2) for any n> ñ.


The proof is based on the Birman-Schwinger argument which, in view of (<ref>), reads

    z∈σ_p (H_α )  ⇔  ∃  n∈ : Γ _n(z)=0  ,
 cf. <cit.>.  Note that, given n∈
the function z↦Γ_n (z), z ∈{ : √(z-n^2)> 0} has the unique
zero at z=ξ_α +n^2, i.e.

    Γ_n (ξ_α +n^2) = 0 .

Finally, it follows, for example, from <cit.> that the corresponding eigenfunctions takes the form 𝒢 (ϵ_l)∗χ_n δ. This completes the proof.




§ SURFACE IMPURITY 
 

We define  a finite smooth parameterized surface  Σ⊂Ω being  a graph of the map U∋ q=(q_1, q_2) ↦ x (q) ∈Ω. 
The surface element can be calculated by means of the standard formula
dΣ = |∂_q_1x (q)×∂_q_1x (q) |dq.
Additionally we assume that  Σ∩ I = ∅.
Furthermore, let n : Σ→^3 stand for the unit normal vector (with an arbitrary orientation) 
and ∂_n denote the normal derivative defined by vector n.
Relying on the Sobolev theorem we state that the trace map W^1,2(Ω ) ∋ψ↦ψ |_Σ∈  L^2 (Σ ) constitutes
a bounded operator;  we set the notation (· , · )_Σ for the
scalar product in L^2 (Σ ).
Given β∈∖{0} we define the following boundary conditions: suppose
that  ψ∈  C (Ω ) ∩ C^1 (Ω∖Σ  )
satisfies

    ∂_n ^+ ψ|_Σ - ∂_n ^- ψ|_Σ= -βψ|_Σ ,

where  the partial derivatives contributing  to the above expression
are defined as the positive, resp., negative limits on Σ 
and signs are understood with respect
to direction of n.
 
   
  D_3)  We say that a function ψ belongs to the set D̆⊂
 W^2,2_loc (Ω∖ (I∪Σ )) if Δψ∈ L^2 (Ω ), ψ|_∂Ω=0 and the limiting equations (<ref>) and (<ref>)   are satisfied.
   
  D_4) Define operator which  for f∈D̆ acts as  -Δ f (x) if x∈Ω∖ (I∪Σ)
   and let H_α, β : D (H_α, β) → L^2 (Ω ) stand for its closure.




To figure out the resolvent of H_α , β we  define the operator acting from
to L^2 (Ω  ) to L^2(Σ  ) as R_α, Σ(z)f= (R_α(z)f )|_L^2(Σ ). Furthermore, we introduce the operator from L^2 (Σ ) to L^2 (Ω )
defined by R_α, Σ (z)f = 𝒢_α∗ f δ, where 𝒢_α stands
for kernel of (<ref>). Finally, we define  R_α, ΣΣ (z) :  L^2 (Σ ) → L^2 (Σ )
by R_α, ΣΣ (z) f= (R_α, Σ (z)f)|_Σ. In view of
(<ref>) the latter takes  the following form

    R_α, ΣΣ (z)= R_ΣΣ(z) +∑_n=1^∞Γ_n (z)^-1  (w_n (z̅ ), · )_Σw_n(z) ,

where w_n(z):= ω_n(z)|_Σ and R_ΣΣ(z) :  L^2 (Σ )→ L^2 (Σ )
stands for the bilateral embedding of R(z).

Following the strategy developed in <cit.> we define the set Z ⊂ρ (H_α ) such that z belongs to Z if the operators

    (I-βR_α , ΣΣ(z))^-1 ,  and   (I-βR_α , ΣΣ(z̅))^-1
 acting from L^2(Σ ) to L^2 (Σ)
   exist and are bounded. 
   Our aim is to show that
   
    Z≠∅ .

  Therefore we auxiliary define the quadratic below bounded  form
   
    ∫_Ω |ψ |^2dx - β∫_Σ| ψ |_Σ|^2dΣ ,  ψ∈ W^1,2_0 (Ω ) .
  Let H̆_β stand for the operator associated to the above form in the sense of the first representation theorem, cf. <cit.>. Following the arguments from <cit.>
 we conclude  that I-βR_ΣΣ(z) : L^2 (Σ)→ L^2 (Σ) defines  the Birman–Schwinger operator for H̆_β.  Using Thm. 2.2 of <cit.> one obtains
 
    z ∈ρ (H̆_β )  ⇔   0∈ρ (I-βR_ΣΣ(z)) .

 In the following we are interested in negative spectral parameter and thus we
  assume z=-λ where λ >0.
Since the spectrum of H̆_β is lower bounded we conclude

    0∈ρ (I-βR_ΣΣ(-λ) ) ,

for λ large enough.

Next step is to find a bound for the second component contributing to (<ref>). In fact, it can be majorized by
   
    ∑_n=1^∞| Γ_n (-λ)^-1| w_n (-λ )_Σ^2 ≤ C
        ∑_n=1^∞w_n (-λ )_Σ^2 ,

   where we applied the uniform bound  | Γ_n (-λ)^-1| ≤ C, cf. (<ref>).
   Using the large argument expansion, cf. <cit.>,
   
    K_0 (z)∼√(π/2z) ^-z

 we get the estimate

    |w_n (-λ ,x)| ≤ C 1/λ^1/4^-r_min(n^2+λ )^1/2  for   λ→∞ ,

where r_min= min _x∈Σ |x|. This implies that  the norm of
the second component of  (<ref>) behave as o(λ^-1). Combining this result with
(<ref>) we conclude that 0∈ρ (I-βR_α, ΣΣ (-λ )) for λ sufficiently large which shows that (<ref>) holds.

To realize the strategy of <cit.> we observe that the embedding operator
τ ^∗ :  L^2 (Σ )→ W^-1,2(Ω ) acting as  τ ^∗ f= f∗δ is bounded and, moreover,

    Ran τ ^∗∩ L^2 (Ω ) = {0} .

Suppose that z∈ Z. Using  (<ref>) together with Thm. 2.1 of <cit.> we conclude that the expression
 
    R_α, β(z)= R_α (z)+  R_α, Σ (z)(I-
      βR_α , ΣΣ(z))^-1 R_α, Σ (z) .

defines the resolvent of self adjoint operator.

   We have
 
    R_α, β(z)= (H_α , β - z)^-1 .

 

To show the statement we repeat the strategy applied in the proof of Theorem <ref>.
Operator H_α, β is defined as the self adjoint  extension of -Δ |_C^∞_0 (Ω∖ (I ∪Σ ))
determined by imposing boundary conditions (<ref>) and (<ref>). The idea is show that
any function from the domain D (H_α, β) satisfies (<ref>) and (<ref>). Since the proof can be done
by the mimicking the arguments from the proof of Theorem <ref> we omit further details.


Furthermore, repeating the arguments from the proof of Theorem <ref> we state that

    σ_ess (H_α , β) = [1 ,∞ ) .



Notation.  In the following we will be interested in the spectral
asymptotic for small  |Σ |. Therefore, we introduce an appropriate scaling with respect to a point x_0 ∈Σ.
Namely,  for small positive  parameter δ we define Σ_δ as the graph of U ∋ q ↦ x_δ (q) ∈Ω where

    x_δ (q):= δ x(q)-δ x_0+x_0 .

For example a sphere of radius R  originated at x_0  turns to  the sphere of radius δ R
after scaling.   Note that equivalence |∂_q_1x_δ (q)×∂_q_1x_δ (q) |=δ^2
|∂_q_1x(q)×∂_q_1x (q) | implies the scaling  of the surface area |Σ_δ |= δ^2 |Σ |.

 

§ PRELIMINARY RESULTS FOR THE ANALYSIS OF POLES

 The Birman-Schwinger argument relates the eigenvalues of H_α, β and zeros
 of I-βR_α , ΣΣ(z) determined by the condition
 (I-βR_α , ΣΣ(z))≠{0}.
 To recover resonances we show that R_α , ΣΣ (z) has a second sheet continuation
 R_α , ΣΣ^𝐼𝐼 (z) and the statement
 
    (I-βR^𝐼𝐼_α , ΣΣ(z))≠{0}

 holds for certain  z∈_-.


 §.§ Analytic continuation of R_α , ΣΣ(z)

We start with the analysis of the first component  of
R_α , ΣΣ(z) determined  by R_ΣΣ(z), cf. (<ref>).
Since R_ΣΣ(z) is defined by means of the embedding of
  kernel 𝒢 (z), see (<ref>),
 the following lemma will be useful for further discussion.
 
For any  k∈ the function  𝒢(z) admits the second sheet continuation
𝒢^𝐼𝐼(z) through J_k:= (k^2, (k+1)^2))
to an open set Π_k ⊂_- and ∂Π_k ∩ =J_k. Moreover,
𝒢^𝐼𝐼(z) takes the form



    𝒢^𝐼𝐼 (z; x,x', x_3, x_3') = 1/2π∑_n=1^∞ Z_0 (i√(z-n^2) |x-x'|)χ_n (x_3) χ_n (x'_3) ,

where

    Z_0 (i√(z-n^2)ρ ) =
         {[                      K_0 (-i√(z-n^2)ρ ),                               for    n>k; K_0 (-i√(z-n^2)ρ )+iπ I_0 (i√(z-n^2)ρ ),                           for    n≤ k  , ].

  and I_0 (·) standardly denotes  the Bessel function.


  The proof is based  on edge-of-the-wedge theorem, i.e. our aim is to establish the convergence
  
    𝒢(λ+i 0) = 𝒢^𝐼𝐼(λ-i 0) ,

  for λ∈ J_k.  In fact, it suffices to show that the analogous formula holds for Z_0 and z=λ± i 0. 

  Assume first that n>k. Then √(λ -n^2 ± i0)= √(λ -n^2 ) since √(λ -n^2 ) >0.
  Furthermore,  the  function  K_0 (·) is analytic in the upper half-plane, consequently,
  we have K_0 (-i√(λ -n^2± i0)ρ )
  = K_0 (√(n^2 -λ)ρ ).

  Assume now that n ≤ k. Then √(λ -n^2 ± i0)= ±√(λ -n^2 )∈ which implies  
    K_0 (-i√(λ -n^2 + i0)ρ )=
      K_0 (-i√(λ -n^2 )ρ ) .


On the other hand, using the analytic continuation formulae
 
    K_0 (z ^mπ i )=K_0 (z) - i mπ I_0 (z)   and   I_0  (z ^mπ i ) = I_0 (z) ,
 for   m∈, we get 
    Z_0 (√(λ -n^2 - i0)ρ  )    =   
       K_0 (i√(λ -n^2 )ρ )+ i π  I_0 (-i√(λ -n^2 )ρ )
        
       =     K_0 (-i√(λ -n^2 )ρ ) .
 This completes the proof.






The above lemma provides  the second sheet continuation of R (z) as well as  R_ΣΣ(z);
the latter is defined
as the bilateral embedding of R^𝐼𝐼 (z) to L^2 (Σ ).
 
  Note that for each k∈ the analytic continuation of 𝒢 (·) through J_k leads to
  different branches. Therefore, we have to keep in mind that the analytic continuation of 𝒢 (·)
  is k-dependent.

In the next lemma we show that the operator R^𝐼𝐼_Σ_δΣ_δ(z
) is bounded and derive the operator norm  asymptotics if  δ→ 0.

 
  Assume that k∈  and λ∈ J_k.  Let z=λ -iε, where ε is a small positive number. Operator R^𝐼𝐼_Σ_δΣ_δ (z) is bounded and its norm admits the asymptotics
  
    R^𝐼𝐼_Σ_δΣ_δ (z)   =o(1) ,

where the error term is understood with respect to δ.


To estimate the kernel of R^𝐼𝐼_Σ_δΣ_δ (z) we use  (<ref>), i.e.

    𝒢^𝐼𝐼_Σ_δΣ_δ(z; ρ , x_3, x_3') = 1/2π(∑ _n=1^∞ K_0 (κ_n (z)ρ )χ_n (x_3)χ_n (x_3')
    . 
    .+∑ _n≤ k  I_0 (-κ_n (z)ρ )χ_n (x_3)χ_n (x_3') ) ,

where 
    ρ =|x-x'| ,  and    x ,x'∈Σ_δ .

First, we consider (<ref>).
The expression  
    |I_0 (-κ_n (z)ρ )χ_n (· )χ_n (· )|
 is bounded.
Therefore the operator defined by the kernel (<ref>) is also bounded and the corresponding operator norm in
L^2 (Σ_δ ) behaves as |Σ_δ |^2 =𝒪(δ^4 ). 

The analysis of the term (<ref>) is more involving because it consists of infinite number of components.
The asymptotics:

    K_0 (κ_n (z)ρ )-K_0 (n ρ ) = ln√(1-z/n^2)(  1+𝒪(ρ ))

implies

    ∑_n=1^∞| (K_0 (κ_n (z)ρ )-K_0 (n ρ ))χ_n (x_3)χ_n (x_3 ') |= C+ 𝒪(ρ ) ;

remind that κ_n (z)  is defined by (<ref>).
To estimate ∑_n=1^∞ K_0 (n ρ )χ_n (x_3)χ_n (x_3 ') we borrow the idea
from <cit.> and use <cit.> to get

    ∑_n=1^∞ K_0 (nρ )cos (na )    =   π/2√(ρ^2+a^2)+1/2(lnρ/4π - ψ(1)) 
           +
      π/2∑ _n=1^∞( 1/√((2nπ +a)^2+ρ ^2)-1/2n π)
           + π/2∑ _n=1^∞( 1/√((2nπ -a)^2+ρ^2)-1/2n π) .

For x,x'∈Σ_δ the terms (<ref>) and (<ref>) can be majorized by
C(∑_n=1^∞1/n^2), i.e. by a uniform constant.
Consequently, using the above estimates  together with the equivalence
sin a sin b =1/2( cos(a-b)-cos(a+b)) we get after
straightforward calculations

    | ∑_n=1^∞ K_0 (κ_n (z)ρ )χ_n (x_3)χ_n (x_3 ')   | ≤ C(1/|x-x'|+
      ln |x-x'|) ;

the singular terms in the above estimates come from (<ref>).
Let us analyze the left hand side of (<ref>). First, we  consider the component 𝒫(x,x'):=1/|x-x'| which
gives

    ∫_Σ_δ𝒫 (x,x') dΣ_δ = (𝒫∗δ_Σ_δ )(x) =
    ∫_Σ_δ1/|x -x' |dΣ_δ .

To conclude the desired convergence we employ the concept
of generalized Kato measure. Namely, since the Dirac delta on Σ_δ
defines Kato measure we obtain

    sup _x∈Σ_δ∫_Σ_δ𝒫 (x,x')  dΣ_δ  = o(1) ,

where the right hand side asymptotics is understood in the sense of convergence with respect  to δ.
Employing the Schur argument we conclude that the norm  of the integral operator  with the kernel
𝒫 (x,x') acting from L^2 (Σ_δ ) to L^2 (Σ_δ ) behaves as o(1).
The term  ln |x-x'| contributing to (<ref>) can be estimated in the analogous way.


To recover the second sheet continuation of R_α, ΣΣ (·)
it remains to construct the analytic extensions of  ω_n (z) and Γ _n (z), cf. (<ref>).
  Given n∈ the functions
ω_n (z) and Γ_n (z) admit the second sheet continuations ω^𝐼𝐼_n (z) and Γ^𝐼𝐼_n (z)
to Π_k through J_k= (k^2, (k+1)^2), k∈ defined by

    ω^𝐼𝐼_n (z; x ):= 1/2πZ_0 (i√(z-n^2 )|x|)χ_n (x_3) ,

where Z_0 is determined by  (<ref>), and


    Γ^𝐼𝐼_n (z)=
         {[       1/2π(2πα -ψ (1)+ln√(z-n^2)/2i),                            for    n>k; 1/2π(2πα -ψ (1)+ln√(z-n^2)/2i- π i ),                        for    n≤ k  . ].



 The construction of ω_n^𝐼𝐼(z) can be obtained mimicking the arguments from the proof of Lemma <ref>.

 To get Γ^𝐼𝐼(z) we first assume k<n and z=λ± iε, λ∈ (k^2, (k+1)^2).   Then
ln√(λ-n^2± i 0)/i= ln√(n^2 -λ) and, consequently, Γ_n(λ +i0)=Γ^𝐼𝐼_n(λ -i0). 

Assume now that n≤ k. Then we have λ -n^2 >0 and
ln√(λ -n^2 ± i0 )/i= ln√(λ -n^2 ± i0 )∓π/2 i 

which implies

    Γ_n(λ +i0)=Γ^𝐼𝐼_n(λ -i0) =
     1/2π(2πα -ψ (1)+ln√(λ -n^2)  - π/2 i ) .

This, in view of edge-of-the-wedge
theorem,  completes the proof.


 

Henceforth, we assume  that ϵ_n ≠ k^2 for any k,n∈. Suppose z=λ-iε, where ε is a small non-negative number and λ∈ J_k.
 At most one eigenvalue ϵ_l can exist in the interval J_k.  Assuming that z ∈ (Π_k ∪ J_k )∖ϵ_l
we define the analytic functions
z↦Γ_n ^𝐼𝐼(z)^-1 for

n∈. Then the second sheet continuation of the resolvent
takes the form

    R^𝐼𝐼 _α ,ΣΣ (z)= R^𝐼𝐼 _ΣΣ (z)+ ∑_n=1^∞Γ_n ^𝐼𝐼(z)^-1(w^𝐼𝐼_n (z̅ ), · )_Σ
      w^𝐼𝐼_n (z )

for z ∈ (Π_k ∪ J_k )∖ϵ_l.


Notation. In the following we will avoid the superscript 𝐼𝐼 keeping in mind that
all quantities
depending on z  are defined for   second sheet continuation if z <0 which admits infinitely many
 branches Π_k, k∈.




Assume that ϵ_l ∈ J_k. Having  in mind latter purposes we define

    A_l (z):= ∑_n≠ lΓ_n (z)^-1 (w_n (z̅), · )_Σ_δw_n (z) ,

for z∈ (Π_k ∪ J_k)∖ϵ_l. The following lemma states the operator
norm asymptotics.
 
 Operator A_l(z) : L^2 (Σ_δ ) → L^2 (Σ_δ ) is bounded and the operator
 norm satisfies
 
    A_l (z)≤ C |Σ_δ | .

 
 
 Suppose that z=λ - iε. We derive   the estimates
 
    |(A_l (z)f,f)_Σ_δ|    ≤   ( ∑_n≠ l  |Γ_n(z)^-1| w_n (z)_Σ_δ^2 ) f_Σ_δ^2
     ≤
           C  ( ∑_n≠ l w_n (z)_Σ_δ^2 ) f_Σ_δ^2 ;

to obtain (<ref>)
we use  (<ref>).
Now our aim is to show

    ∑_n≠ l w_n (z)_Σ_δ^2  ≤ C  |Σ_δ | .

To find a bound for the left hand side of (<ref>) we analyse first
the behaviour of w_n (z) for large n and z∈ (Π_k ∪ J_k)∖ϵ_l.
For this aim we employ (<ref>) and (<ref>).
Note that for  n>k
function w_n (z) admits the representation:

    w_n (z,x)= 1/2π K_0 (κ_n (z)|x|) χ_n (x_3) ,

where x∈Σ_δ.
Using again the large argument expansion  (<ref>) 
and the fact that
(-i √(z-n^2))∼  n we get the estimate

    |w_n (z,x)| ≤ C^-r_minn ,

where r_min= min _x∈Σ_δ |x|. This implies

    ∑_n >  k , n≠ l w_n (z)_Σ_δ^2  ≤ C|Σ_δ |= 𝒪 (δ^2 ) .

On the other hand for n≤ k function w_n (z) consists of K_0 and I_0, see (<ref>). Both functions are continuous
on Σ_δ and therefore w_n^2_Σ_δ≤ C|Σ_δ |. Since the number of such components is finite, in view of
(<ref>),
we come to (<ref>) which completes the proof.
 

 

§ COMPLEX POLES OF RESOVENT
 

Assume that ϵ_l ∈ J_k. Suppose that δ is sufficiently small.  It follows from Lemmae <ref> and <ref> that the operators
 I-βR_Σ_δΣ_δ (z) and I-β A_l (z) acting in L^2 (Σ_δ )
 are invertible for z∈ (Π_k ∪ J_k)∖ϵ_l and it makes sense to introduce auxiliary notation
  
    G_Σ_δ (z):= (I-βR_Σ_δΣ_δ (z) )^-1 .

  Since the norm of G_Σ_δ(z) A_l (z) : L^2 (Σ_δ) → L^2 (Σ_δ) tends to 0 if δ→ 0 therefore
  the operator I +β G_Σ_δ(z) A_l (z) is invertible as well.
  
 The following theorem “transfer” the analysis of resonances from the operator equation
  to the complex valued function equation.
  
   Suppose  ϵ_l ∈ J_k and assume that z∈ (Π_k ∪ J_k )∖ϵ_l.  Then the condition
   
    (
       I-βR_α , Σ_δΣ_δ (z))  ≠{0}
  is equivalent to
   
    Γ_l (z) +β ( w_l (z̅), T_l (z)w_l (z) ) _Σ_δ=0 ,

where
   
    T_l (z) :=  (I -β G_Σ_δ(z) A_l (z))^-1G_Σ_δ(z) .

    
   
   The strategy of the proof is partially based on the idea borrowed from <cit.>.
The following equivalences
 
    I-   βR_α ,  Σ_δΣ_δ(z)    =   
       =   (I- βR_Σ_δΣ_δ(z) )
       (I -β G_Σ_δ(z) A_l (z). 
          .   -βΓ_l (z)^-1(w_l (z̅ ), · )_Σ_δ G_Σ_δ(z)w_l (z) )
       =    (I- βR_Σ_δΣ_δ(z) )(I  - β G_Σ_δ(z) A_l (z)) ×
          [ I  -βΓ_l (z)^-1(w_l (z̅ ), · )_Σ_δ T_l (z)w_l (z) ]
        ,

 show that (<ref>) is equivalent to
 
    [ I  -βΓ_l (z)^-1(w_l (z̅ ), · )_Σ_δ T_l (z)w_l (z) ] ≠{0 } .

 The above condition is formulated for  a rank one operator and, consequently, it is equivalent
 to (<ref>).
   
   

   Theorem <ref> shows that the problem of complex poles of resolvent R_α, β(z) can be shifted
   to  the problem of the roots analysis of
   
    η_l (z,δ )=0  ,  where   η_l (z,δ ):=Γ_l (z) - βϑ_l (z,δ ) ,

   and
   
    ϑ_l (z,δ ):=   (  w_l (z̅), T_l (z)w_l (z) ) _Σ_δ .

   The further discussion is devoted to figuring out roots
   of  (<ref>). In the following we apply the expansion  (1+A)^-1=(1-A+A^2-A^3...) valid if A<1.
   Taking -βR_Σ_δΣ_δ (z) 
   as A we get
 
    G_Σ_δ (z)=(I- βR_Σ_δΣ_δ (z))^-1=I+ R̆(z) ,  R̆(z):=∑_n=1(βR_Σ_δΣ_δ (z))^n  .

Expanding the analogous sum for -β G_Σ_δ (z) A_l (z) one obtains
 
    (I- β G_Σ_δ (z) A_l (z))^-1= I+   β A_l (z) + βR̆(z)A_l (z) +...

   In view of Lemmae <ref> and <ref> the norm of R_Σ_δΣ_δ (z)A_l (z) behaves
   as o(1)  A_l (z)_Σ_δ for δ→ 0 and the same asymptotics holds for the operator norm of
   R̆(z)A_l (z).
    The further terms in (<ref>) are of smaller order with respect
   to δ.
    Consequently,  applying again (<ref>) we conclude that T_l (z) admits the following expansion
   
    T_l (z)= I+β A_l (z)+ R̆(z)+...

   Using the above statements we can formulate the main result.
   
     Suppose that ϵ_l ∈ J_k and consider the function η_l (z,δ ) : Π_k ∪ J_k × [0, δ_0)→,
     where δ_0>0,  defined by  (<ref>). Then the equation
     
    η_l (z, δ ) =0 ,

     possesses a solution which is determined by the function  δ↦ z(δ )∈ with the following asymptotics
   
    z_l(δ )= ϵ_l+μ_l (δ ) ,    |μ_l(δ )|= o(1) .

Moreover, the lowest order term of μ_l (· ) takes the form

    μ_l(δ ) =        4πξ_αβ{w_l (ϵ_l )^2_Σ_δ.
    
          .
    +β∑_n≠ l Γ_n (ϵ_l )^-1 |(w_l (ϵ_l), w_n (ϵ_l))_Σ_δ|^2
    .
    
          .  +   (w_l (ϵ_l ),R̆(ϵ_l) w_l (ϵ_l ) )_Σ_δ} .

   
   
   Note that  z↦η_l (z,δ ), cf. (<ref>), is analytic and
   η_l (ϵ_l , 0) = 0. Using  (<ref>) 
   one obtains 
    . d Γ _n (z)/dz|_z=ϵ_n =1/4πξ_α <0  ,    n∈.

Combining this with
 
    . ∂ϑ_n(z,δ )/∂ z|_z=ϵ_l, δ =0  = 0 ,

   
   we get
     .∂η_l /∂ z|_δ=0= 1/4πξ_α≠ 0.
     In view of the Implicit Function Theorem we conclude that the equation (<ref>)
      admits a unique solution
     which a continuous function of δ↦ z_l(δ) and z_l (δ )= ϵ_l +o(1).
     To reconstruct asymptotics of z(· )
     first we expand Γ_l (z)  into the Taylor sum
     
    Γ_l (z)= 1/4πξ_α (z-ϵ_l) +𝒪((z-ϵ_l)^2) .

     Then the spectral equation (<ref>) reads
     
    z=ϵ_l + 4 πξ_αβϑ_l (z,δ )+𝒪((z-ϵ_l)^2) .

      Now we expand ϑ_l (z,δ ). Using  (<ref>) and (<ref>)
   we reconstruct its  first order term which reads
      
    {w_l (ϵ_l )^2_Σ_δ
    
    +β∑_n≠ l Γ_n (ϵ_l )^-1 |(w_l (ϵ_l), w_n (ϵ_l))_Σ_δ|^2
    .
    
          .
       + (w_l (ϵ_l ),R̆(ϵ_l) w_l (ϵ_l ) )_Σ_δ} .

       Applying the asymptotics
     z_l(ϵ_l)=ϵ_l +o(δ ) and the fact that ϑ_l (· ,·  ) is analytic with respect to
     complex variable we get formula for μ(·).
   


   

 §.§ Analysis of imaginary part of the pole 


Since the imaginary component of resonance pole has a physical meaning
we dedicate to this problem a special discussion.
The information on the lowest order term of the pole imaginary component is contained in    (<ref>) and (<ref>). On the other  hand, note that
only the components subscripted by n≤ k admit a non-zero imaginary parts. Therefore

    ( 4πξ_αβ( β∑_n≤ k  Γ_n (ϵ_l )^-1 |(w_l (ϵ_l), w_n (ϵ_l))_Σ_δ|^2
    . .
          . .+ (w_l (ϵ_l ),R̆(ϵ_l) w_l (ϵ_l ) )_Σ_δ) )  .

determines the lowest order term of μ (δ ).

 

Sign and asymptotics of μ (δ )  with respect to Σ_δ .
Recall that ϵ_l ∈ J_k. First we analyse (<ref>) and for this aim we define

    ι_l,n: = 1/2π( 2πα +ln√(ϵ_l -n^2)/2-ψ(1)) ,

for n≤ k.
Relying on (<ref>) we get

    Γ_l (ϵ_l )^-1 = 1/ι_l,n^2 +(1/2)^2(ι_l,n +1/2i )

if  n≤ k. Consequently,  formula (<ref>) is equivalent to

    4πξ_αβ^2
    ∑_n ≤ k 1/21/ι_l,n^2 +(1/2)^2
      |(w_l (ϵ_l), w_n (ϵ_l))_Σ_δ|^2 .

The above expression is negative because  ξ_α <0.
Moreover, since both w_l (ϵ_l) and w_l (ϵ_n) are continuous in Ω∖ I
we have |(w_l (ϵ_l), w_n (ϵ_l))_Σ_δ|^2∼ |Σ_δ |^2. This means that  (<ref>) behaves  as 𝒪(|Σ_δ|^2). To recover the asymptotics of (<ref>) we
restrict ourselves  to the lowest order term of R̆(z), cf. (<ref>), namely

    υ_l :=  4πξ_αβ ^2 (w_l (ϵ_l ),R_Σ_δΣ _δ(ϵ_l) w_l (ϵ_l ) )_Σ_δ .

Using analytic continuation formulae   (<ref>) and employing the small argument expansion, cf. <cit.>,




    K_0 (z) ∼  -ln z ,

where -π < z <π states the plane cut for the logarithmic
 function,
one gets

    υ_l ∼πξ_αβ ^2
    ∑_n≤  k ( ∫_Σ_δ w_l (ϵ_l )χ_n )^2 =𝒪(|Σ_δ |^2 ) .


One can easily see that υ_l <0.
Summing up the above discussion we can formulate the following conclusion.


The resonance pole takes the form z_l (δ )=ϵ_l+μ (δ ) with the lowest order of μ (δ ) given by

    πξ_αβ ^2   ∑_n ≤ k (  2/ι_l,n^2 +(1/2)^2
      |(w_l (ϵ_l), w_n (ϵ_l))_Σ_δ|^2 +
     ( ∫_Σ_δ w_l (ϵ_l )χ_n )^2 )  .

It follows from the above formula that
μ (δ )<0 and
the asymptotics 
    μ (δ )= 𝒪(|Σ_δ |^2)
 holds. Moreover, the lowest order
of μ (δ ) is independent of sign of β.








Note that for the special geometrical cases the embedded eigenvalues can survive
after introducing Σ_δ since the "perturbed" eigenfunctions are not affected
by presence of Σ_δ.
Let us consider
 
    Π_l := {x∈Ω :  x=(x, π/l ) ,    l∈

 and assume that Σ_δ⊂Π_l. Then w_ml (z) = 0 for each m∈ and, consequently
 ϑ_ml (z,δ ) = 0, cf. (<ref>). This implies the following statement.


  Suppose that Σ⊂Π_l. Then
  for all m∈ the numbers ϵ_ml remain the embedded eigenvalues of H_α , β.





 §.§ Acknowledgements


The  author  thanks  the  referees  for  reading  the  paper  carefully, removing errors and
recommending
various improvements in exposition. 

The work was  supported
 by the project DEC-2013/11/B/ST1/03067 of the Polish National Science Centre.



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