Spectral analysis of sheared nanoribbons
Philippe Briet Hamza Abdou Soimadou David Krejcirik










Spectral analysis of sheared nanoribbons

    Philippe Briet,^a  Hamza Abdou Soimadou^a  
and  David Krejčiřík^b
    



  a)
Aix-Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France;
briet@univ-tln.fr, abdou-soimadou-hamza@etud.univ-tln.fr

  b)
Department of Mathematics, Faculty of Nuclear Sciences and 
Physical Engineering, Czech Technical University in Prague, 
Trojanova 13, 12000 Prague 2, Czechia;
david.krejcirik@fjfi.cvut.cz




26 July 2018
=================================================================================================================================================================================================================================================================================================================================================================



 


We investigate the spectrum of the Dirichlet Laplacian in a unbounded strip
subject to a new deformation of “shearing”:
the strip is built by translating a segment oriented 
in a constant direction along an unbounded curve in the plane.
We locate the essential spectrum under the hypothesis that
the projection of the tangent vector of the curve 
to the direction of the segment admits a (possibly unbounded) limit at infinity  
and state sufficient conditions which guarantee the existence 
of discrete eigenvalues.
We justify the optimality of these conditions by establishing
a spectral stability in opposite regimes.
In particular, Hardy-type inequalities are derived 
in the regime of repulsive shearing.



  Keywords:
sheared strips, quantum waveguides, Hardy inequality, Dirichlet Laplacian.

  MSC (2010):
Primary: 35R45; 81Q10; Secondary: 35J10, 58J50, 78A50.










§ INTRODUCTION
 


With advances in nanofabrication and measurement science,
waveguide-shaped nanostructures have reached the point at which
the electron transport can be strongly affected by quantum effects.
Among the most influential theoretical results,
let us quote the existence of quantum bound states due to bending
in curved strips, 
firstly observed by Exner and Šeba <cit.> in 1989.
The pioneering paper has been followed by a huge number 
of works demonstrating the robustness of the effect 
in various geometric settings including higher dimensions,
and the research field is still active these days
(see <cit.> for a recent paper
with a brief overview in the introduction).

In 2008 Ekholm, Kovařík and one of the present authors <cit.>
observed that the geometric deformation of twisting 
has a quite opposite effect on the energy spectrum 
of an electron confined to three-dimensional tubes,
for it creates an effectively repulsive interaction 
(see <cit.> for an overview of 
the two reciprocal effects). 
More specifically, twisting the tube locally gives rise 
to Hardy-type inequalities and a stability of quantum transport,
the effect becomes stronger in globally twisted tubes <cit.>
and in extreme situations it may even annihilate 
the essential spectrum completely <cit.>
(see also <cit.>).
The repulsive effect remains effective even under modification 
of the boundary conditions <cit.>.

The objective of this paper is to introduce a new, two-dimensional model 
exhibiting a previously unobserved geometric effect of shearing.
Mathematically, it is reminiscent of the effect of twisting 
in the three-dimensional tubes 
(in some aspects it also recalls 
the geometric setting of curved wedges studied in <cit.>), 
but the lower dimensional simplicity
enables us to get an insight into analogous problems left open in <cit.>
and actually provide a complete spectral picture now. 
The richness of the toy model is reflected in covering 
very distinct regimes, ranging from purely essential 
to purely discrete spectra or a combination of both.
We believe that the present study will stimulate 
further interest in sheared nanostructures. 

The model that we consider in this paper is characterised
by a positive number d (the transverse width of the waveguide) 
and a differentiable function f:→ 
(the boundary profile of the waveguide).
The waveguide Ω is introduced as the set of points in ^2
delimited by the curve x ↦ (x,f(x)) 
and its vertical translation x ↦ (x,f(x)+d), 
namely (see Figure <ref>), 


    Ω := { 
      (x,y) ∈^2 : f(x) < y < f(x)+d
      }
      .



We stress that the geometry of Ω 
differs from the curved strips intensively studied
in the literature for the last thirty years
(see <cit.> for a review).
In the latter case, the segment (0,d) is translated along 
the curve x ↦ (x,f(x)) with respect to its normal vector field
(so that the waveguide is delimited by two parallel curves),
while in the present model the translation is with respect to 
the constant basis vector in the y-direction.






Clearly, it is rather the derivative f' that determines the shear	
deformation of the straight waveguide Ω_0 := × (0,d).
Our standing assumption is that the derivative f' 
is locally bounded and that it admits a (possibly infinite) 
limit at infinity:


    lim_|x| →∞ f'(x) =: β∈∪{±∞} 
      .


If β is finite,
we often denote the deficit 


    (x):=f'(x)-β ,


but notice that (x) is not necessarily small 
for finite x.
	
We consider an effectively free electron constrained to
the nanostructure Ω by hard-wall boundaries.
Disregarding physical constants, the quantum Hamiltonian of 
the system can be identified with the Dirichlet Laplacian 
-Δ_D^Ω in (Ω).
The spectrum of the straight waveguide Ω_0 
(which can be identified with f'=0 in our model)
is well known; by separation of variables,
one easily conclude that
σ(-Δ_D^Ω_0) = [(π/d)^2,∞)
and the spectrum is purely absolutely continuous.
The main spectral properties of -Δ_D^Ω
under the shear deformation obtained in this paper
are summarised in the following theorems.

First, we locate the possible range of energies of propagating states.

[Essential spectrum]
Let f' ∈ L_loc^∞() satisfy (<ref>).
Then


    σ_ess(-Δ_D^Ω) = [E_1(β),∞)
       ,         
      E_1(β) := (1+β^2) (π/d)^2  .





If β = ±∞ (see Figure <ref>), 
the result means σ_ess(-Δ_D^Ω) = ∅,
so the spectrum is purely discrete 
and there are no propagating states. 
In this case, the distance of points x∈Ω
to the boundary ∂Ω tends to zero as |x| →∞,
so Ω is a quasi-bounded domain ( Remark <ref>).
This phenomenon is reminiscent of three-dimensional waveguides
with asymptotically diverging twisting 
<cit.>.






Our next concern is about the possible existence or absence
of discrete eigenvalues below E_1(β) if β is finite.
The following theorem together with Theorem <ref>
provides a sufficient condition for the existence.


[Attractive shearing]
Let f' ∈ L_loc^∞() 
be such that f'^2 - β^2 ∈ L^1().
Then either of the following conditions



  (i)
∫_ (f'^2(x) - β^2)   x < 0
   ;

  (ii)
∫_ (f'^2(x) - β^2)   x = 0,
f' is not constant 
and f”∈ L_loc^1()  ;


implies 


    infσ(-Δ_D^Ω) < E_1(β)
       .





The theorem indeed implies the existence of discrete spectra
under the additional hypothesis (<ref>),
because the inequality (<ref>)
and the fact that, due to Theorem <ref>,
the essential spectrum of -Δ_D^Ω starts by E_1(β)
ensure that the lowest point in the spectrum of -Δ_D^Ω
is an isolated eigenvalue of finite multiplicity.
Theorem <ref> is an analogy of <cit.>,
where it was shown that a local “slow down” of twisting 
in periodically twisted tubes generates discrete eigenvalues
(see also <cit.>).
It is not clear from our proof whether the extra regularity
assumption in the condition (ii) is necessary. 

Recalling (<ref>),
notice that f'^2-β^2 = ^2 + 2 β,
so a necessary condition to satisfy either of 
the conditions (i) or (ii) of Theorem <ref>
is that the function β is not non-negative.  
This motivates us to use the terminology that 
the shear is attractive (respectively, repulsive)
if β is non-positive (respectively, non-negative)
and  is non-trivial.
In analogy with the conjectures stated in <cit.>,
it is expected that the discrete spectrum of -Δ_D^Ω
is empty provided that the shear is either repulsive 
or possibly attractive but ^2 ≫ -β. 
The latter is confirmed in the following special setting
(see Figure <ref>).


[Strong shearing]
Let f'(x) = β + α(x), where 
α, β∈ and :→ 
is a function such that
⊂ [0,1] and c_1 ≤(x) ≤ c_2
for all x∈[0,1] with some positive constants c_1,c_2.
Then there exists a positive number α_0 such that,
for all |α| ≥α_0,


    σ(-Δ_D^Ω) = [E_1(β),∞)
       .





Finally, we establish Hardy-type inequalities 
in the case of repulsive shear.


[Repulsive shearing]
Let f' ∈ L_loc^∞().
If β≥ 0 and  is non-trivial, 
then there exists a positive constant c such that the inequality


    -Δ_D^Ω - E_1(β) ≥c/1+x^2


holds in the sense of quadratic forms in (Ω).



Consequently, if the shear satisfies (<ref>)
in addition to the repulsiveness,
then (<ref>) and Theorem <ref> imply 
that the stability result (<ref>) holds,
so in particular there are no discrete eigenvalues.
Moreover, the spectrum is stable against small perturbations
(the smallness being quantified by the Hardy weight
on the right-hand side of (<ref>)).
From Theorem <ref> it is also clear that α
does not need to be large if αβ≥ 0
in the setting of Theorem <ref>.
 
The organisation of the paper is as follows.
In Section <ref> we introduce natural curvilinear coordinates
to parameterise Ω and express the Dirichlet Laplacian in them.
Hardy-type inequalities (and in particular Theorem <ref>) 
are established in Section <ref>. 
The essential spectrum (Theorem <ref>) 
is l in Section <ref>.
The sufficient conditions of Theorem <ref>
(which in particular imply the existence of discrete eigenvalues
in the regime (<ref>))
are established in Section <ref>.
Finally, Theorem <ref> is proved in Section <ref>.




§ PRELIMINARIES



Our strategy to deal with the Laplacian in the sheared geometry Ω
is to express it in natural curvilinear coordinates. 
It employs the identification 
Ω = ℒ(Ω_0),
where ℒ:^2→^2 is the shear mapping defined by


    ℒ(s,t) := (s,f(s)+t)
      .


In this paper we consistently use the notations s and t
to denote the “longitudinal” and “transversal” 
variables in the straight waveguide Ω_0, respectively. 
The corresponding metric reads


    G := ∇ℒ· (∇ℒ)^T = 
      [ 1+f'^2     f';     f'      1 ]
      ,     
      |G| := (G) = 1
      ,


where the dot denotes the scalar product in ^2.
Recall our standing assumption 	that f' ∈ L_loc^∞().
It follows that ℒ:Ω_0 →Ω is a local diffeomorphism.
In fact, it is a global diffeomorphism, because ℒ is injective.
In this way, we have identified Ω 
with the Riemannian manifold (Ω_0,G).

The Dirichlet Laplacian -Δ_D^Ω is introduced standardly
as the self-adjoint operator in the Hilbert space (Ω)
associated with the quadratic form Q_D^Ω[u] := ∫_Ω |∇ u|^2,
Q_D^Ω := H_0^1(Ω). 
Employing the diffeomorphism ℒ:Ω_0 →Ω,
we have the unitary transform

    𝒰 : (Ω) →(Ω_0) :
      {u ↦ u ∘ℒ} .

Consequently, -Δ_D^Ω is unitarily equivalent 
(and therefore isospectral)
to the operator H := 𝒰(-Δ_D^Ω)𝒰^-1
in the Hilbert space (Ω_0).
By definition, the operator H is associated with the quadratic form

    h[ψ] := Q_D^Ω[𝒰^-1ψ] 
       ,      h := 𝒰 Q_D^Ω .

Occasionally, we shall emphasise the dependence of h and H on f'
by writing f' as the subscript, 
 h=h_f' and H=H_f'.

Given u ∈ C_0^∞(Ω), a core of Q_D^Ω,
it follows that ψ := u ∘ℒ
is also compactly supported and ψ∈ H_0^1(Ω_0).
Then it is straightforward to verify that


    h[ψ] = ∂_sψ-f'∂_tψ^2 + ∂_tψ^2
       ,


where · denotes the norm of (Ω_0)
and, with an abuse of notation, we denote by the same symbol f'
the function f' ⊗ 1 on ×(0,d).
We have the elementary bounds


    δ∂_sψ^2 
      + (1-δ/1-δf'_L^∞(ψ)^2) 
      ∂_tψ^2
      
    ≤ h[ψ] ≤
      2 ∂_sψ^2 
      + (1 + 2 f'_L^∞(ψ)^2) ∂_tψ^2


valid for every δ∈ (0,1).	
The domain of h coincides with the closure of the set 
of the functions ψ specified above	
with respect to the graph topology of h.
By a standard mollification argument using (<ref>), 	
it follows that C_0^∞(Ω_0) is a core of h. 	
Moreover, it follows from (<ref>) that
if f' is bounded, then the graph topology of h
is equivalent to the topology of H^1(Ω_0),
and therefore h = H_0^1(Ω_0) in this case.
In general (and in particular if (<ref>) holds with β = ±∞),
however, the domain of h is not necessarily equal to H_0^1(Ω_0).  
Let us summarise the preceding observations into the following proposition.



Let f' ∈ L_loc^∞().
Then C_0^∞(Ω_0) is a core of h.
If moreover f' ∈ L^∞(), then h = H_0^1(Ω_0). 



In a distributional sense, 
we have H = - (∂_s-f'∂_t)^2 - ∂_t^2,
but we shall use neither this formula nor any information
on the operator domain of H.
Our spectral analysis of H will be exclusively based on
the associated quadratic form (<ref>).
Notice that the structure of H is similar to 
twisted three-dimensional tubes <cit.>
as well as curved two-dimensional wedges <cit.>.	




§ HARDY-TYPE INEQUALITIES



In this section we prove Theorem <ref>.
The strategy is to first establish a “local” Hardy-type inequality,
for which the Hardy weight might not be everywhere positive,
and then “smeared it out” to the “global” inequality (<ref>)
with help of a variant of the classical one-dimensional Hardy inequality.
Throughout this section, we assume that β∈
and ∈ L_loc^∞().	 

Let 

    E_1 := (π/d)^2
              χ_1(t) := √(2/d) sin(E_1 t)

denote respectively the lowest eigenvalue and the corresponding
normalised eigenfunction of the Dirichlet Laplacian in ((0,d)).  
By the variational definition of E_1,
we have the Poincaré-type inequality


    ∀χ∈ H_0^1((0,d))  ,     ∫_0^d |χ'(t)|^2   t ≥ E_1 ∫_0^d |χ(t)|^2   t
       .


Notice that E_1 = E_1(0), where the latter is introduced in (<ref>).
Neglecting the first term on the right-hand side of (<ref>)
and using (<ref>) for the second term
together with Fubini's theorem, we immediately get the lower bound


    H ≥ E_1
       ,


which is independent of f.	

The main ingredient in our approach is the following valuable lemma 
which reveals a finer structure of the form h.
With an abuse of notation, we use the same symbol χ_1
for the function 1 ⊗χ_1 on × (0,d),
and similarly for its derivative χ_1',
while  also denotes the function 
⊗ 1 on × (0,d). 


[Ground-state decomposition]
For every ψ∈ C_0^∞(Ω_0), we have


    h[ψ] - E_1(β) ψ^2
      =     ∂_sψ
      - ∂_t ψ - βχ_1∂_t(χ_1^-1ψ) ^2 
      + χ_1∂_t(χ_1^-1ψ)^2
      
        + ∫_Ω_0β (s)[
      E_1 + (χ_1'(t)/χ_1(t))^2
      ]
      |ψ(s,t)|^2   s   t
      .







Let ϕ∈ C_0^∞(Ω_0) be defined 	
by the decomposition ψ(s,t) = χ_1(t) ϕ(s,t).
Writing 2(ϕ∂_tϕ) = ∂_t |ϕ|^2,
integrating by parts and using the differential equation
that χ_1 satisfies, we have 

    ∂_tψ^2 = χ_1'ϕ^2 + χ_1∂_tϕ^2
      + 2(χ_1'ϕ,χ_1∂_tϕ)
      = χ_1∂_tϕ^2 + E_1 ψ^2
       ,

where (·,·) denotes the inner product of (Ω_0).
Similarly,


    ∂_sψ-f'∂_tψ^2
      =    ∂_sψ-∂_tψ - βχ_1∂_tϕ^2 
      + β^2 χ_1'ϕ^2
      
       - 2β (χ_1∂_sϕ
      -χ_1∂_tϕ - χ_1'ϕ - βχ_1∂_tϕ,
      χ_1'ϕ)
      
    
      =    ∂_sψ-∂_tψ - βχ_1∂_tϕ^2 
      
        + β E_1 (χ_1ϕ,χ_1ϕ)
      +β (χ_1'ϕ,χ_1'ϕ)
      + β^2 E_1 (χ_1ϕ,χ_1ϕ)
      
    
      =    ∂_sψ-∂_tψ - βχ_1∂_tϕ^2 
      
        + β E_1 (ψ,ψ)
      + β (χ_1'ϕ,χ_1'ϕ)
      + β^2 E_1 ψ^2 
       ,


where the second equality follows by integrations by parts 
in the cross term.
Summing up the results of the two computations
and subtracting E_1(β) ψ^2, 
we arrive at the desired identity.



From now on, let us assume that the shear is repulsive, 
 β≥ 0 and ≠ 0.
If β≠0, then Lemma <ref> immediately
gives the following local Hardy-type inequality


    H - E_1(β) ≥β[
      E_1 + (χ_1'/χ_1)^2
      ]
      .


Recall that E_1(β) corresponds to the threshold
of the essential spectrum if (<ref>) holds ( Theorem <ref>),
so (<ref>) in particular ensures that 
there is no (discrete) spectrum below E_1(β)
(even if β=0). 
Note that the Hardy weight on the right-hand side of (<ref>)
diverges on ∂Ω_0.
The terminology “local” comes from the fact that 
the Hardy weight might not be everywhere positive 
( if β is compactly supported).	

In order to obtain a non-trivial non-negative lower bound
including the case β=0, 
we have to exploit the positive terms
that we have neglected when coming from (<ref>) to (<ref>).	
To do so, let I ⊂ be any bounded open interval 
and set Ω_0^I := I×(0,d).
In the Hilbert space (Ω_0^I) let us consider
the quadratic form

    q_I[ψ] 
         := 
      ∂_sψ
      - ∂_t ψ 
      - βχ_1∂_t(χ_1^-1ψ) _(Ω_0^I)^2 
      + χ_1∂_t(χ_1^-1ψ)_(Ω_0^I)^2
       , 
     q_I
         := {ψ↾Ω_0^I:
      ψ∈ H_0^1(Ω_0)} .

The form corresponds to the sum of the two first terms 
on the right-hand side of (<ref>)
with the integrals restricted to Ω_0^I.




The form q_I is closed. 



First of all, notice that 
the form q_I is obviously closed if β=0
(one can proceed as in (<ref>)). 
In general, given ψ∈ q_I,
we employ the estimates


    δ∂_sψ - ∂_t ψ_(Ω_0^I)^2
      + (1-δ/1-δ β^2) 
      χ_1∂_t(χ_1^-1ψ)_(Ω_0^I)^2
      
    ≤ q_I[ψ] ≤
      2 ∂_sψ - ∂_t ψ_(Ω_0^I)^2
      + (1+2β^2)χ_1∂_t(χ_1^-1ψ)_(Ω_0^I)^2


valid for every δ∈ (0,1)  
and the identity 
(easily checked by employing the density of 
C_0^∞(Ω_0) in H_0^1(Ω_0))


    χ_1∂_t(χ_1^-1ψ)_(Ω_0^I)^2
      = ∂_tψ_(Ω_0^I)^2 - E_1 ψ_(Ω_0^I)^2
       .


Instead of the latter, we could also use the fact that
χ_1^-1ψ∈(Ω_0) whenever ψ∈ H_0^1(Ω_0),
 <cit.>.
Using in addition the aforementioned fact that q_I is closed if β=0,
we conclude that q_I is closed for any value β∈.



Consequently, the form q_I
is associated to a self-adjoint operator Q_I with compact resolvent.
Let us denote by λ_I its lowest eigenvalue, ,


    λ_I := inf_[ψ≠0]ψ∈ q_Iq_I[ψ]/ ψ_(Ω_0^I)^2 .


The eigenvalue λ_I is obviously non-negative.
The following lemma shows that λ_I is actually positive
whenever  is non-trivial on I. 



λ_I=0 if, and only if, =0 on I. 





If ↾ I = 0, 
then the function ψ(s,t):=χ_1(t)
minimises (<ref>) with λ_I=0.
Conversely, let us assume that λ_I=0
and denote by ψ_1 the corresponding eigenfunction of Q_I,
 the minimiser of (<ref>).  
It follows from (<ref>) together with (<ref>) that 	


    ∂_sψ_1
      - ∂_t ψ_1 
      - βχ_1∂_t(χ_1^-1ψ_1) _(Ω_0^I)^2    = 0
       ,
      
    ∂_tψ_1_(Ω_0^I)^2
      -E_1ψ_1_(Ω_0^I)^2    = 0
       .


Writing ψ_1(s,t) = φ(s)χ_1(t) + ϕ(s,t),
where

    ∫_0^d χ_1(t)ϕ(s,t)   t = 0

for almost every s ∈ I, we have 

    ∂_tψ_1_(Ω_0^I)^2
      -E_1ψ_1_(Ω_0^I)^2  
      = ∂_tϕ_(Ω_0^I)^2
      -E_1ϕ_(Ω_0^I)^2  
      ≥ (E_2-E_1) ϕ_(Ω_0^I)^2
       ,

where E_2=4E_1 
is the second eigenvalue of the Dirichlet Laplacian in ((0,d)).
Since the gap E_2-E_1 is positive, 
it follows from the second identity in (<ref>) that ϕ=0.
Plugging now the separated-eigenfunction Ansatz
ψ_1(s,t) = φ(s)χ_1(t)
into the first identity in (<ref>)
and integrating by parts with respect to t in the cross term, 
we obtain

    0 = ∂_sψ_1-∂_tψ_1_(Ω_0^I)^2 
      = φ'χ_1_(Ω_0^I)^2 
      + φχ_1'_(Ω_0^I)^2 
       .

Consequently, φ is necessarily constant 
and _(I)=0.  



Using (<ref>), we can improve (<ref>) 
to the following local Hardy-type inequality


    H - E_1(β) ≥β[
      E_1 + (χ_1'/χ_1)^2
      ]
      + λ_I   1_I×(0,d) ,


where 1_M denotes the characteristic function of a set M.
The inequality is valid with any bounded interval I ⊂,
but recall that λ_I is positive if, and only if, I is chosen in 
such a way that  is not identically equal to zero on I,
 Lemma <ref>. 

The passage from the local Hardy-type inequality (<ref>)
to the global Hardy-type inequality of Theorem <ref>
will be enabled by means
of the following crucial lemma, 
which is essentially a variant of the classical
one-dimensional Hardy inequality.



Let s_0 ∈ and
ψ∈ C_0^∞(Ω_0∖{s_0}×(0,d)). 
Then


    ∂_sψ
      - ∂_t ψ - βχ_1∂_t(χ_1^-1ψ)^2 
      + χ_1∂_t(χ_1^-1ψ)^2
      ≥1/4(1+β^2)∫_Ω_0|ψ(s,t)|^2/(s-s_0)^2  s   t 
       .






 
Denote ρ(s,t):=(s-s_0)^-1.
Given any real number α, we write


    ∂_sψ
      - ∂_t ψ - βχ_1∂_t(χ_1^-1ψ)
      - αρψ^2
    
      =    ∂_sψ
      - ∂_t ψ - βχ_1∂_t(χ_1^-1ψ)^2
      + α^2 ρψ^2
      - 2 α (
      ∂_sψ
      - ∂_t ψ - βχ_1∂_t(χ_1^-1ψ),
      ρψ
      )
      
    
      =    ∂_sψ
      - ∂_t ψ - βχ_1∂_t(χ_1^-1ψ)^2
      + (α^2-α) ρψ^2
      + 2αβ(χ_1∂_t(χ_1^-1ψ),ρψ)
      
    ≤    ∂_sψ
      - ∂_t ψ - βχ_1∂_t(χ_1^-1ψ)^2
      + (α^2-α+α^2β^2) ρψ^2
      + χ_1∂_t(χ_1^-1ψ)^2	
       ,


where the second equality follows by integrations by parts
and the estimate is due to the Schwarz inequality.
Since the first line is non-negative, 
we arrive at the desired inequality
by choosing the optimal α := [2(1+β^2)]^-1.



Now we are in a position to prove the “global” Hardy inequality 
of Theorem <ref>.


Let ψ∈ C_0^∞(Ω_0).
First of all, since  is assumed to be non-trivial, 
there exists a real number s_0 and positive b
such that  restricted to the interval 
I := (s_0-b,s_0+b) is non-trivial.
By Lemma <ref>, it follows that λ_I is positive.
Then the local Hardy-type inequality (<ref>) implies
the estimate


    h[ψ] - E_1(β) ψ^2
      ≥λ_I ψ_(Ω_0^I)^2
       .



Second,
let η∈ C^∞() be such that 0 ≤η≤ 1,
η=0 in a neighbourhood of s_0 and η=1 outside I. 
Let us denote by the same symbol η the function 
η⊗ 1 on × (0,d),
and similarly for its derivative η'. 
Writing ψ = ηψ + (1-η)ψ, 
we have


    ∫_Ω_0|ψ(s,t)|^2/1+(s-s_0)^2  s   t
      ≤ 2 ∫_Ω_0|(ηψ)(s,t)|^2/(s-s_0)^2  s   t 
      + 2 ∫_Ω_0 
      |((1-η)ψ)(s,t)|^2  s   t 
       .


Since ηψ satisfies the hypothesis of Lemma <ref>, 
we have 


    ∫_Ω_0|(ηψ)(s,t)|^2/(s-s_0)^2  s   t 
      
       ≤ 4 (1+β^2) (
      ∂_s(ηψ)
      - η∂_t ψ - ηβχ_1∂_t(χ_1^-1ψ)^2 
      + ηχ_1∂_t(χ_1^-1ψ)^2
      )
      
       ≤ 8 (1+β^2) ∂_sψ
      - ∂_t ψ - βχ_1∂_t(χ_1^-1ψ)^2 
      + 8 (1+β^2) η'ψ
      + 4 (1+β^2)  χ_1∂_t(χ_1^-1ψ)^2	
      
       ≤ 8 (1+β^2) (h[ψ]-E_1(β)ψ^2)
      + 8 (1+β^2) η'_∞ψ_(Ω_0^I)^2


where η'_∞ denotes the supremum norm of η'.
Here the last inequality employs Lemma <ref>
(recall that we assume that the shear is repulsive). 
At the same time,

    ∫_Ω_0 
      |((1-η)ψ)(s,t)|^2  s   t 
      ≤ψ_(Ω_0^I)^2
       .

From (<ref>) we therefore deduce	


    h[ψ] - E_1(β) ψ^2
      ≥1/16(1+β^2)∫_Ω_0|ψ(s,t)|^2/1+(s-s_0)^2  s   t
      - (
      η'_∞^2 + 1/8(1+β^2))
      ψ_(Ω_0^I)^2
       .



Finally, interpolating between (<ref>) and (<ref>),
we obtain 


    h[ψ] - E_1(β) ψ^2
      ≥    δ/16(1+β^2)∫_Ω_0|ψ(s,t)|^2/1+(s-s_0)^2  s   t
      
        + [
      (1-δ) λ_I
      - δη'_∞^2 - δ/8(1+β^2)]
      ψ_(Ω_0^I)^2


for every real δ.
Choosing (positive) δ in such a way
that the square bracket vanishes,
we arrive at the global Hardy-type inequality

    h[ψ] - E_1(β) ψ^2
      ≥ 
      c' ∫_Ω_0|ψ(s,t)|^2/1+(s-s_0)^2  s   t

with

    c' := 
      λ_I/
      16(1+β^2) (λ_I + η'_∞^2) + 2

From this inequality we also deduce 


    h[ψ] - E_1(β) ψ^2
      ≥ 
      c ∫_Ω_0|ψ(s,t)|^2/1+s^2  s   t


with

    c := c' inf_s∈1+s^2/1+(s-s_0)^2 ,

where the infimum is positive.
The desired inequality (<ref>) follows
by the unitary equivalence between H and -Δ_D^Ω
together with the fact that the the coordinates s and x
are equivalent through this transformation,  (<ref>).





The present proof does not employ the presence 	of the first term
on the right-hand side of (<ref>).
The inequality (<ref>)
can be consequently improved to

    H - E_1(β) ≥δ β(s) 
      [
      E_1 + (χ_1'(t)/χ_1(t))^2
      ]
      + (1-δ)  c/1+s^2

with any δ∈ [0,1].







§ LOCATION OF THE ESSENTIAL SPECTRUM
 


In this section we prove Theorem <ref>.	
Since the technical approaches for finite and infinite β
are quite different, we accordingly split the section 
into two subsections.
In either case, we always assume f' ∈ L_loc^∞(). 



 §.§ Finite limits


We employ the following characterisation of the essential spectrum,
for which we are inspired in <cit.>.



A real number λ belongs to the essential spectrum of H
if, and only if, there exists a sequence 
{ψ_n}_n=1^∞⊂ h such that
the following three conditions hold:



  (i)
ψ_n = 1 for every n ≥ 1,

  (ii)
(H-λ)ψ_n → 0
as n→∞
in the norm of the dual space ( h)^*.

  (iii)
ψ_n ⊂Ω_0 ∖ (-n,n)×(0,d)
for every n ≥ 1.







By the general Weyl criterion modified to quadratic forms 
( <cit.> or <cit.>
where the latter contains a proof),
λ belongs to the essential spectrum of H
if, and only if, there exists a sequence 
{ϕ_n}_n=1^∞⊂ h such that (i) and (ii) hold
but (iii) is replaced by 



  (iii')
ϕ_n → 0 as n →∞ weakly in (Ω_0).


The sequence {ψ_n}_n=1^∞
satisfying (i) and (iii) is clearly weakly converging to zero. 
Hence, one implication of the lemma is obvious.
Conversely, let us assume that there exists 
a sequence {ϕ_n}_n=1^∞⊂ h
satisfying (i), (ii) and (iii')
and let us construct from it a sequence {ψ_n}_n=1^∞⊂ h   
satisfying (i), (ii) and (iii).
Since C_0^∞(Ω_0) is the form core of H
( Proposition <ref>),
we may assume that {ϕ_n}_n∈⊂ C_0^∞(Ω_0).

Let us denote the norms of h and ( h)^* by
·_+1 and ·_-1, respectively.
One has ·_± 1 = (H+1)^± 1/2·.
By writing


    ϕ_n = (H+1)^-1 (H-λ) ϕ_n + (λ+1) (H+1)^-1ϕ_n


and using (ii),
we see that the sequence {ϕ_n}_n=1^∞ 
is bounded in h.

Let η∈ C^∞() be such that 0 ≤η≤ 1,
η=0 on [-1,1] and η=1 on ∖(-2,2). 
For every k ≥ 1, we set η_k(s) := η(s/k)
and we keep the same notation for the function 
η_k ⊗ 1 on × (0,d),
and similarly for its derivatives η_k' and η_k”.
Clearly, η_k ⊂Ω_0 ∖ (-k,k)×(0,d). 
For every k ≥ 1, the operator (1-η_k)(H+1)^-1
is compact in (Ω_0).
By virtue of the weak convergence (iii'), 
it follows that, for every k ≥ 1, 
(1-η_k)(H+1)^-1ϕ_n → 0 as n →∞
in (Ω_0).
Then there exists a subsequence 
{ϕ_n_k}_k=1^∞ of {ϕ_n}_n=1^∞ 
such that (1-η_k)(H+1)^-1ϕ_n_k→ 0 as k →∞
in (Ω_0). 
Consequently,
the identity (<ref>) together with (ii)
implies that (1-η_k)ϕ_n_k→ 0 as k →∞ in (Ω_0).
It follows that η_k ϕ_n_k can be normalised
for all sufficiently large k. More specifically, 
redefining the subsequence if necessary, we may assume that
η_k ϕ_n_k≥ 1/2 for all k ≥ 1.
We set

    ψ_k := η_k ϕ_n_k/η_k ϕ_n_k

and observe that {ψ_k}_k=1^∞ satisfies (i) and (iii). 

It remains to verify (ii).
To this purpose, we notice that, using the duality, (ii) means


    (H-λ)ϕ_n_-1
      = sup_[φ≠0]φ∈ h|h(φ,ϕ_n)-λ(φ,ϕ_n)|/φ_+1 0
       ,


where φ_+1^2 = h[φ] + φ^2.
By the direct computation employing integrations by parts,
one can check the identity 


    h(φ,η_k ϕ_n_k) - λ (φ,η_k ϕ_n_k)
      
    
      = h(η_kφ,ϕ_n_k) - λ (η_kφ,ϕ_n_k)
      + 2(∂_sφ-f'∂_tφ,η_k'ϕ_n_k)
      + (φ,η_k”ϕ_n_k)


for every test function φ∈ C_0^∞(Ω_0),
a core of h.
Using (<ref>), we have

    sup_[φ≠0]φ∈ C_0^∞(Ω_0)|h(η_kφ,ϕ_n_k)
      -λ(η_kφ,ϕ_n_k)|/φ_+1   ≤sup_[η_kφ≠0]φ∈ C_0^∞(Ω_0)|h(η_kφ,ϕ_n_k)
      -λ(η_kφ,ϕ_n_k)|/η_kφ_+1
       = (H-λ)ϕ_n_k_-1 0
       .

At the same time, using the Schwarz inequality
and the estimates 
∂_sφ-f'∂_tφ^2 ≤ h[φ] 
  ≤φ_+1^2,
we get

    sup_[φ≠0]φ∈ C_0^∞(Ω_0)|2(∂_sφ-f'∂_tφ,η_k'ϕ_n_k)|/φ_+1   ≤η_k'_∞ϕ_n_k ,

where η_k'_∞ denotes the supremum norm of η_k'.
In view of the normalisation (i) 
and since η_k'_∞ = k^-1η'_∞,
we see that also this term tends to zero as k →∞.
Finally, using the Schwarz inequality 
and the estimate φ≤φ_+1,
we have


    sup_[φ≠0]φ∈ C_0^∞(Ω_0)|(φ,η_k”ϕ_n_k)|/φ_+1≤η_k”_∞ϕ_n_k
      = k^-2η”_∞ 0
       .


Summing up, we have just checked 
(H-λ)(η_k ϕ_n_k)_-1→ 0 as k→∞.
Recalling η_k ϕ_n_k≥ 1/2,
the desired property (ii) for {ψ_k}_k=1^∞ follows. 



Using this lemma, we immediately arrive at the following
“decomposition principle” (saying that the essential spectrum
is determined by the behaviour at infinity only).



If (<ref>) holds with β∈, then 
σ_ess(H_β+) 
  = σ_ess(H_β).





Let λ∈σ_ess(H_β).
By Lemma <ref>, there exists a sequence 
{ψ_n}_n=1^∞ satisfying the properties (i)–(iii) 
with H being replaced by H_β.
One easily checks the identity


    h_β+(φ,ψ_n)
      = h_β(φ,ψ_n)
      - (∂_sφ-β∂_tφ,∂_tψ_n)
      - (∂_tφ,∂_sψ_n-(β+)∂_tψ_n)


for every test function 
φ∈ h_β
  = H_0^1(Ω_0)
  =  h_β+
( Proposition <ref>).
Now we proceed similarly as in the end of the proof of Lemma <ref>.
Using the estimates 
∂_sφ-β∂_tφ^2
  ≤ h[φ]
  ≤φ_+1
and the Schwarz inequality,
we have

    sup_[φ≠0]φ∈ H_0^1(Ω_0)|(∂_sφ-β∂_tφ,∂_tψ_n)|/φ_+1   ≤∂_tψ_n≤_L^∞(∖(-n,n))∂_tψ_n
      0
       .

Recall that {ψ_n}_n=1^∞ is bounded in H_0^1(Ω_0),
 (<ref>) and Proposition <ref>.
Similarly, using ∂_tφ≤φ_+1
and the Schwarz inequality, we obtain

    sup_[φ≠0]φ∈ H_0^1(Ω_0)|(∂_tφ,∂_sψ_n-(β+)∂_tψ_n)|/φ_+1   ≤∂_sψ_n 
      + β+_∞∂_tψ_n
       ≤_L^∞(∖(-n,n)) 
      (∂_sψ_n 
      + β+_∞∂_tψ_n)
      
       
      0
       .

Consequently, since (H_β-λ)ψ_n_-1→ 0
as n→∞ due to (ii), 
it follows from (<ref>) and (<ref>) that also
(H_β+-λ)ψ_n_-1→ 0
as n →∞. 
Hence λ∈σ_ess(H_β+).

The opposite inclusion 
σ_ess(H_β+) 
  ⊂σ_ess(H_β)
is proved analogously.



It remains to determine the (essential) spectrum 
for the constant shear.



One has
σ(H_β) 
  = σ_ess(H_β) 
  = [E_1(β),∞).





By performing the partial Fourier transform 
in the longitudinal variable s ∈,
one has the unitary equivalence

    H_β = -(∂_s-β∂_t)^2 - ∂_t^2 
      ≅ -(iξ-β∂_t)^2 - ∂_t^2 
       ,

where ξ∈ is the dual variable in the Fourier image.
Consequently, 


    σ(H_β) = ⋃_ξ∈σ(T_β(ξ))
       ,


where, for each fixed ξ∈, 
T_β(ξ) is the operator in ((0,d))
that acts as -(iξ-β∂_t)^2 - ∂_t^2
and satisfies Dirichlet boundary conditions.
The spectral problem for T_β(ξ) can be solved explicitly. 
Alternatively,
one can proceed by “completing the square” 
and “gauging out a constant magnetic field” as follows:

    -(iξ-β∂_t)^2 - ∂_t^2
         = (1+β^2) (- ∂_t^2)
      + 2iξβ∂_t
      + ξ^2
      
       = - (1+β^2) (∂_t-i ξβ/1+β^2)^2
      - ξ^2β^2/1+β^2 + ξ^2	
      
       = - (1+β^2) (∂_t-i ξβ/1+β^2)^2
      + ξ^2/1+β^2
       = 
      e^iFt[ - (1+β^2) ∂_t^2 + ξ^2/1+β^2]
      e^-iFt ,

where 

    F := ξβ/1+β^2 .

Consequently, 


    σ(T_β(ξ)) = {
      (1+β^2) E_n + ξ^2/1+β^2}_n=1^∞ ,


where E_n = n^2 E_1 denotes the n^th eigenvalue
of the Dirichlet Laplacian in ((0,d)).
As a consequence of (<ref>) and (<ref>),
we get the desired claim. 



For finite β,
Theorem <ref> follows as a direct consequence
of Propositions <ref> and <ref>.



 §.§ Infinite limits


Let us now prove Theorem <ref> in the case β = ±∞.
Here the proof is based on the following purely geometric fact.



Let f' ∈ L_loc^∞() satisfy (<ref>)
with β = ±∞. Then


    lim sup_[x∈Ω]|x|→∞ 
      |B_1(x) ∩Ω| = 0
       ,


where B_1(x) denotes the disk of radius 1 
centred at x∈^2
and |·| stands for the Lebesgue measure.   



Recall the diffeomorphism ℒ:Ω_0→Ω given by (<ref>).
For every number η∈,
let Σ(η) denote the set of points 
intersecting the boundary ∂Ω
with the straight horizontal half-line (0,∞) ×{η}.
Since f'(s) →±∞ as |s|→∞, 
it follows that there exists a positive constant s_0
such that f is either strictly increasing or strictly decreasing on (s_0,∞).
Let us consider the former situation, the latter can be treated analogously. 
It follows that there exists a positive constant η_0 
such that, for all η≥η_0,
the set Σ(η) consists of just two points
x_1(η)=ℒ(s_1(η),d) 
and x_2(η)=ℒ(s_2(η),0)
with 1 < s_1(η) < s_2(η).
Let x=ℒ(s,t) be a point in Ω 
such that s is positive  
and the vertical component satisfies the inequality η := f(s)+t ≥ 1 + η_0
(notice that |x| →∞ if, and only if, |s| →∞).
Then we have the crude estimate

    |B_1(x) ∩Ω|
      ≤ 2   |s_2(η+1) - s_1(η-1)|
       .

By the mean value theorem,

    2 = f(s_2(η+1)) - f(s_1(η-1))
      = f'(ξ_η)   |s_2(η+1) - s_1(η-1)|
       ,

where ξ_η∈ (s_1(η-1),s_2(η+1)).
Since s_1(η-1) →∞ as s →∞,
we also have ξ_η→∞,
and therefore |B_1(x) ∩Ω| → 0 as s →∞.
The case s → -∞ can be treated analogously.



By the Berger-Schechter criterion 
( <cit.>),
it follows from (<ref>)
that the embedding W_0^1,2(Ω) ↪ L^2(Ω) is compact.
Consequently, the Dirichlet Laplacian -Δ_D^Ω 
has no essential spectrum if β = ±∞
and Theorem <ref> is proved.



A necessary condition for (<ref>) is that 


    lim_[x∈Ω]|x|→∞(x,∂Ω) = 0
       ,


which means that Ω is a quasi-bounded domain 
(<cit.>) in the regime β = ±∞.







§ EXISTENCE OF DISCRETE SPECTRUM



In view of Theorem <ref>,
the conclusion (<ref>) of Theorem <ref> guarantees
that any of the conditions (i) or (ii) implies 
that -Δ_D^Ω possesses
at least one isolated eigenvalue of finite multiplicity below E_1(β). 
Let us establish these sufficient conditions.



Given any positive n,
let φ_n:→[0,1] be the continuous function
such that φ_n=1 on [-n,n],
φ_n=0 on ∖ (-2n,2n)
and linear in the remaining intervals.
Set 

    ψ_n(s,t) := φ_n(s) χ_1(t)

and observe that ψ_n ∈(h) ∩ H_0^1(Ω_0).
Moreover, the identity of Lemma <ref> 
remains valid for ψ_n by approximation. 
Indeed, it is only important to notice that
χ_1^-1ψ_n ∈(Ω_0)
as a consequence of <cit.>.
Consequently,


    h[ψ_n] - E_1(β) ψ_n^2
         =  ∂_sψ_n
      - ∂_t ψ_n^2 
      + ∫_Ω_0β (s)[
      E_1 + (χ_1'(t)/χ_1(t))^2
      ]
      |ψ_n(s,t)|^2   s   t
      
       = φ_n'_()^2
      + E_1 ∫_[(s)^2 + 2 β(s)]
      |φ_n(s)|^2   s
       ,


where the second equality follows by the special form of ψ_n
(the cross term vanishes due to an integration by parts with respect to t)
and the formula χ_1'_((0,d))^2=E_1
together with the normalisation of χ_1.
Noticing that φ_n'_()^2=2n^-1
and using the dominated convergence theorem,
we get

    h[ψ_n] - E_1(β) ψ_n^2
      
      E_1 ∫_[(s)^2 + 2 β(s)]
       s
       ,

where the right-hand side is negative by the hypothesis (i).
Consequently, there exists a positive n such that
h[ψ_n] - E_1(β) ψ_n^2 < 0,
so the desired result (<ref>) follows by 
the variational characterisation of the lowest point
in the spectrum of H. 

Now assume (ii), in which case the shifted form
h_1[ψ_n] := 
  h[ψ_n] - E_1(β) ψ_n^2 
converges to zero as n →∞.
Then we modify the test function ψ_n 
by adding a small perturbation:

    ψ_n,δ(s,t) := ψ_n(s,t) + δ ϕ(s,t)
              ϕ(s,t) := ξ(s)   t  χ_1(t)
       ,

where δ is a real number 
and ξ∈ C_0^∞() is a real-valued function
to be determined later.
Writing 


    lim_n→∞ h_1[ψ_n,δ] 
         = lim_n→∞(
      h_1[ψ_n] + 2δ  h_1(ψ_n,ϕ) + δ^2   h_1[ϕ]
      )
      
       = 2δlim_n→∞ h_1(ψ_n,ϕ) + δ^2   h_1[ϕ]
       ,


it is enough to show, in order to establish (<ref>), 
that the limit of h_1(ψ_n,ϕ) as n→∞ is non-zero
for a suitable choice of ξ.
Indeed, it then suffices to choose δ sufficiently small
and of suitable sign to make the second line of (<ref>) negative,
and subsequently choose n sufficiently large to make 
the whole expression h_1[ψ_n,δ] negative. 
Employing Lemma <ref> 
and the fact that φ_n=1 on ξ 
for all sufficiently large n, 
we have


    lim_n→∞ h_1(ψ_n,ϕ) 
      =     ∫_Ω_0{χ_1(t) χ_1'(t)   [(s)^2ξ(s) + β(s) ξ(s)]
      - χ_1(t) χ_1'(t) t  (s)ξ'(s)
      
        + χ'(t)^2 t   [(s)^2ξ(s)+β(s) ξ(s)]
      + χ_1(t)^2 t   E_1 β(s) ξ(s)
      }  s   t
      
    
      =     1/2∫_{(s)ξ'(s) + E_1   d   [(s)^2+2β(s)]  ξ(s)
      } s
      
    
      =     1/2∫_{
      -'(s) + E_1   d   [(s)^2+2β(s)] 
      }ξ(s)
        s
       .


Here the last equality follows by an integration by parts
employing the extra hypothesis that the derivative ' 
exists as a locally integrable function. 
Now let assume, by contradiction, that the last integral
equals zero for all possible choices of ξ.
Then  must solve the differential equation
-'(s) + E_1   d   [(s)^2+2β(s)] = 0,
which admits the explicit one-parametric class of solutions

    _c(s) = 2β/c   e^-2 E_1 d β s-1 ,      
      c ∈ .
 
The solutions _c for non-zero c are not admissible
because f'^2-β is either a positive (if c>0) 
or negative (if c<0) function, so its integral cannot be equal to zero 
(for positive c the function _c additionally admits
a singularity at s=-(2 E_1 d β)^-1log c),
while _0 is excluded by the requirement that f' is not constant. 
Hence, there exists a compactly supported ξ such that
the limit of h_1(ψ_n,ϕ) as n→∞ is non-zero
and the proof is concluded by the argument described below (<ref>). 







§ STRONG SHEARING



In this last section we prove Theorem <ref>.
Accordingly, let us assume that 
the shear admits the special form f'(x) = β + α(x),
where α, β∈ and :→ 
is a function such that
⊂ [0,1] and c_1 ≤(x) ≤ c_2
for all x∈[0,1] with some positive constants c_1,c_2.
We divide the proof into several subsections.



 §.§ Preliminaries


If β is non-negative (respectively, non-positive),
then the desired stability result (<ref>) 
obviously holds for all α≥ 0 (respectively, α≤ 0)
due to the repulsiveness of the shear ( Theorem <ref>).  
On the other hand, the shear becomes attractive 
if β is positive (respectively, negative)
and α is small in absolute value and negative (respectively, positive),
so (<ref>) cannot hold in these cases ( Theorem <ref>).     
The non-trivial part of Theorem <ref> therefore 
consists in the statement that (<ref>) holds again
in the previous regimes provided that α becomes large in absolute value.    

Without loss of generality, it is enough to prove the remaining claim 
for β positive, α negative and the shear function 


    f(x) := 
      β x       x<0  ,
        
    
        β x + α∫_0^x (s)   s
              0 ≤ x ≤ 1  ,
        
    
        β x + α∫_0^1 (s)   s 
              1 < x  .


As above, we define Ω by (<ref>)
and do not highlight the dependence on α.
We use the notation (x,y) ∈^2 
for the Cartesian coordinates in which we describe Ω.



 §.§ Subdomains decomposition


Our strategy is to decompose Ω into a union of three 
open subsets Ω_ext, Ω_ver, Ω_int
and of two connecting one-dimensional interfaces  
Σ_ext, Σ_int 
on which we impose extra Neumann boundary conditions later on. 
The definition of the (α-dependent) sets given below
is best followed by consulting Figure <ref>.





Let A∈^2 be the orthogonal projection of the origin O:=(0,0) ∈^2 
on the half-line {(x,f(x)+d): x ≤ 0} 
and let Ω_ext^- be the unbounded open subdomain of Ω
contained in the left half-plane and delimited by 
the open segment Σ_ext^- connecting O with A.
Similarly, let A' be the orthogonal projection of the point O' := (1,f(1)+d)
on the half-line {(x,f(x)): x ≥ 1} 
and let Ω_ext^+ be the unbounded open subdomain of Ω
contained in the right half-plane and delimited by 
the open segment Σ_ext^+ connecting O' with A'.
Set Ω_ext := Ω_ext^- ∪Ω_ext^+
and Σ_ext := Σ_ext^- ∪Σ_ext^+.

Let x_0 ∈ (0,1) be the unique solution of the equation 
f(x_0)+d = 0 and set B:=(x_0,0).
Similarly, let x_0' ∈ (0,1) be the unique solution of the the equation
f(x_0')=f(1)+d and set B':=(x_0',f(1)+d).
Let us also introduce C:=(0,f(0)+d) and C':=(1,f(1)).
Let Ω_ver^- (respectively, Ω_ver^+)
be the quadrilateral-like subdomain of Ω
determined by the vertices O, A, C and B  
(respectively, O', A', C' and B').
Let Σ_int^- (respectively, Σ_int^+)
be the open segment connecting O with B 
(respectively, O' with B').
Set Ω_ver := Ω_ver^- ∪Ω_ver^+
and Σ_int := Σ_int^- ∪Σ_int^+.

Finally, we define 
Ω_int := 
  Ω∖(Ω_ext∪Ω_ver∪Σ_ext∪Σ_int),
which is the parallelogram-like domain	 determined by the vertices OBO'B'.



 §.§ Neumann bracketing


Notice that the majority of the sets introduced above depend on α,
although it is not explicitly highlighted by the notation.
In particular, the parallelogram-like domain determined by the vertices OCO'C'
(subset of Ω_ver∪Ω_int)
converges in a sense to the half-line {0}× (-∞,0)
as α→ -∞.
Hence it is expected that this set is spectrally negligible in the limit
and the spectrum of -Δ_D^Ω converges as α→ -∞
to the spectrum of the Dirichlet Laplacian in {(x,y)∈Ω:x<0}.

Since we only need to show that the spectral threshold 
of -Δ_D^Ω is bounded from below by E_1(β)
for all sufficiently large α, 
in the following we prove less.
Instead, we impose extra Neumann boundary conditions on
Σ_ext∪Σ_int
( no boundary condition in the form setting)
and show that the spectral threshold of the Laplacian
with combined Dirichlet and Neumann boundary conditions
in the decoupled subsets 
Ω_ext, Ω_ver, Ω_int
is bounded from below by E_1(β).
More specifically, 
by a standard Neumann bracketing argument
( <cit.>),
we have 

    infσ(-Δ_D^Ω) 
      ≥min{infσ(-Δ_DN^Ω_ext),
      infσ(-Δ_DN^Ω_ver),
      infσ(-Δ_DN^Ω_int)
      }
      ,
 
where -Δ_DN^Ω_ext 
(respectively, -Δ_DN^Ω_ver;
respectively, -Δ_DN^Ω_int)
is the operator in (Ω_ext)
(respectively, (Ω_ver); 
respectively, (Ω_int))
that acts as the Laplacian 
and satisfies Neumann boundary conditions on Σ_ext
(respectively, on Σ_ext∪Σ_int;
respectively, on Σ_int)
and Dirichlet boundary conditions 
on the remaining parts of the boundary.
It remains to study the spectral thresholds
of the individual operators. 



 §.§ Spectral threshold of the decomposed subsets





  §.§.§ Exterior sets


Notice that Ω_ext consists of two connected components,
each of them being congruent 
to the half-strip (0,∞)×(0,d/√(1+β^2)).
Consequently, -Δ_DN^Ω_ext is isospectral to
the Laplacian in the half-strip, subject to Neumann boundary
conditions on {0}×(0,d/√(1+β^2))
and Dirichlet boundary conditions on the remaining parts of the boundary.
By a separation of variables, it is straightforward to see that
σ(-Δ_DN^Ω_ext)=[E_1(β),∞).
In particular, the spectral threshold of -Δ_DN^Ω_ext
equals E_1(β) for all α.



  §.§.§ Interior set


Given any y ∈ [f(1)+d,0],
let x_1(y) (respectively, x_2(y)) be the unique solution of the equation
f(x_1(y))=y (respectively, f(x_2(y))+d=y).
Notice that x_1(y) < x_2(y). 
In particular, x_1(0)=0 and x_2(0)=x_0,
while x_1(f(1)+d)=x_0' and x_2(f(1)+d)=f(1)+d.
To estimate the spectral threshold of -Δ_DN^Ω_int,
we write

    ∫_Ω_int |∇ψ(x,y)|^2   x   y
         ≥∫_Ω_int |∂_xψ(x,y)|^2   x   y
      
       ≥∫_Ω_int(π/x_2(y)-x_1(y))^2
      |ψ(x,y)|^2   x   y
       ,

where the second inequality follows by a Poincaré inequality
of the type (<ref>) and Fubini's theorem.
Subtracting the equations that x_1(y) and x_2(y) satisfy,
the mean value theorem yields 
d=f(x_1(y))-f(x_2(y)) = -f'(ξ) [x_2(y)-x_1(y)]
with 0<ξ<x_0<1.
Since f'(x)=β+α(x) ≥β - c_2 |α| for x ∈ [0,1],
it follows that, for all α < β/c_2, 
we have the uniform bound
x_2(y)-x_1(y) ≥ d / (c_2|α|-β).
Consequently, 
infσ(-Δ_DN^Ω_int) →∞ 
as α→ -∞.
In particular, the spectral threshold of -Δ_DN^Ω_int
is greater than E_1(β) for all negative α 
with sufficiently large |α|.



  §.§.§ Verge sets


The decisive set Ω_ver consists of two connected 
components Ω_ver^±.
We consider Ω_ver^-, 
the argument for Ω_ver^+ being analogous.
Let us thus study the spectral threshold of 
the operator -Δ_DN^Ω_ver^- in (Ω_ver^-)
that acts as the Laplacian and satisfies Neumann boundary conditions on
Σ_ext^- ∪Σ_int^- 
and Dirichlet boundary conditions on the remaining parts 
of the boundary ∂Ω_ver^-.
Since the spectrum of this operator is purely discrete,
we are interested in the behaviour of its lowest eigenvalue.

As α→ -∞, the set Ω_ver^-
converges in a sense to the open right triangle T 
determined by the vertices OAC.
More specifically, |Ω_ver^- ∖ T|=O(|α|^-1)
as α→ -∞. 
Using the convergence result <cit.>,
it particularly follows that the lowest eigenvalue of 
-Δ_DN^Ω_ver^-
converges to the lowest eigenvalue λ_1(T) 
of the Laplacian in T,
subject to Neumann boundary conditions on the segment OA
and Dirichlet boundary conditions on the other parts of the boundary. 
It remains to study the latter.

By a trivial-extension argument, 
λ_1(T) is bounded from below by the lowest eigenvalue of 
the Laplacian in the rectangle (0,|OA|) × (0,|AC|),
subject to Neumann boundary conditions on (0,|OA|) ×{0}
and Dirichlet boundary conditions on the other parts of the boundary.
That is, 
λ_1(T) ≥ (1+β^-2)   E_1(β).

Summing up, we have established the result

    infσ(-Δ_DN^Ω_ver)
      λ_1(T) ≥ (1+β^-2)   E_1(β)
       .

Consequently, there exists a negative α_0 such that,
for every α≤α_0, the spectral threshold 
of -Δ_DN^Ω_ver is also bounded
from below by E_1(β).
This concludes the proof of Theorem <ref>.
 






 §.§ Acknowledgment




D.K. was partially supported 
by the GACR grant No. 18-08835S
and by FCT (Portugal)
through project PTDC/MAT-CAL/4334/2014.









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