Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem
Anne-Sophie Bonnet-Ben Dhia Lucas Chesnel Vincent Pagneux
math.NA



 



Trapped modes and reflectionless modes as
  eigenfunctions of the same  spectral problem





Anne-Sophie Bonnet-Ben Dhia^1, Lucas Chesnel^2, Vincent Pagneux^3

0.96


^1 Laboratoire Poems, CNRS/ENSTA/INRIA, Ensta ParisTech, Université Paris-Saclay, 828, Boulevard des Maréchaux, 91762 Palaiseau, France;

^2 INRIA/Centre de mathématiques appliquées, École Polytechnique, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau, France;

^3 Laboratoire d'Acoustique de l'Université du Maine, Av. Olivier Messiaen, 72085 Le Mans, France.

E-mails: ,  , 



– December 30, 2023–









Abstract. We consider the reflection-transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-selfadjoint spectral problem. In order to select ingoing waves on one side of the obstacle and outgoing  waves on the other side, we use complex scalings (or Perfectly Matched Layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of 𝒫𝒯-symmetric problems.


Key words. Waveguides, transmission spectrum, backscattering, trapped modes, complex scaling, 𝒫𝒯 symmetry. 





§ INTRODUCTION


In bounded domains, eigenvalues and corresponding eigenmodes completely determine the solutions to wave equations. In open systems, it is necessary to take into account also complex resonances, sometimes known as leaky modes or quasi normal modes in literature <cit.>.  Complex resonances provide the backbone of phenomena of wave propagation. For example, they indicate if there is a rapid variation of the scattering coefficients. We emphasize that real eigenvalues and complex resonances are intrinsic and universal objects that can be computed using different approaches. Among them, let us mention methods based on complex scaling or Perfectly Matched Layers (PMLs) <cit.> or techniques relying on Hardy space infinite elements <cit.>. 
However complex resonances fail to provide useful information on recurrent questions such as “how large is the backscattered energy?” or “is there good transmission in the system?”. 
Such questions are of tremendous importance in many topics of intense study in wave physics: extraordinary optical transmission <cit.>,  
topological states immune to backscattering  <cit.>,
perfect transmission resonances <cit.>, 
transmission eigenchannels through disordered media <cit.>,
reflectionless metamaterials <cit.> or
metasurfaces <cit.>.

In this article, we define a new complex spectrum to quantify these aspects.  In particular, our spectrum allows one to identify wavenumbers for which there is an incident field such that the backscattered field is evanescent (Figure <ref>). For construction of waveguides where such phenomenon occurs at a prescribed frequency, we refer the reader to <cit.>. Our spectrum also contains usual trapped modes <cit.> called Bound States in the Continuum (BSCs or BICs) in quantum mechanics <cit.>. To compute this new spectrum, we use complex scaling/PMLs techniques in an original way. Our approach is based on the following basic observation: if for an incident wave, the backscattered field is evanescent, then the total field is ingoing in the input lead and outgoing in the output lead. Therefore, changing the sign of the imaginary part of the usual complex scaling/PML in the input lead should allow one to compute reflectionless modes (see the exact definition after (<ref>)). We show that this simple idea, inspired by the results obtained in <cit.> on a 1D 𝒫𝒯-symmetric problem, indeed works. Interestingly, we will observe numerically that complex eigenvalues in this spectrum seem to contain useful information as well, indicating settings where reflection is weak.




To make the presentation as simple as possible, we stick to a simple 2D wave problem in a waveguide with Neumann boundary conditions. We assume that this waveguide contains a penetrable obstacle (Figure <ref>). Everything presented here can be generalized to other types of obstacles, to other kinds of boundary conditions (Dirichlet, ...), to higher dimension (3D) and to more complex geometries (one input lead/several output leads).

The paper is organized as follows. In section <ref>, the governing equations of the scattering problem are presented and the reflectionless case is defined. Section <ref> is a reminder of the technique of complex scaling allowing one to compute  classical complex resonances. In section <ref>, we explain how to use conjugated complex scalings to identify reflectionless frequencies. We provide examples of spectra in section <ref> dedicated to numerics and we give some proofs of the theoretical results in section <ref>. Eventually, section <ref> contains some concluding remarks.





§ SETTING




We are interested in the propagation of waves in a waveguide with a bounded penetrable obstacle (Figure <ref>). We assume that the waveguide coincides with the region :={(x,y)∈^2 | 0<y<1} and that the propagation of waves is governed by the Helmholtz equation with Neumann boundary conditions 

    [ Δ u + k^2γ u            =            0             ;        ∂_y u            =            0      y=0y=1. ]

Here u can represent for instance the acoustic pressure in a compressible fluid, or the transverse component of the displacement field in an elastic solid, or one component of the electromagnetic  field in a dielectric medium.  Δ denotes the  Laplace operator. Moreover k is the wavenumber such that k=/c where c denotes  the  waves velocity and   is the angular frequency, corresponding to a time harmonic dependence in e^-i t. We assume that γ is a positive and bounded function such that γ=1 for |x|≥ L where L>0 is given. In other words, the obstacle is located in the region _L:={(x,y)∈ | |x|<L}. 

Pick k∈(Nπ;(N+1)π), with N∈:={0,1,…}. For n∈, set β_n=√(k^2-n^2π^2). In this work, the square root is such that if z=|z| e^iarg(z) with arg(z)∈[0;2π), then √(z)=√(|z|) e^iarg(z)/2. With this choice, we have m √(z)≥0 for all z∈. The function 

    w_n^±(x,y)=(2|β_n|)^-1/2e^± iβ_n xφ_n(y),

with φ_0(y)=1, φ_n(y)=√(2)cos(nπ y) for n≥1, solves Problem (<ref>) without obstacle (γ≡1). For n=0,…,N, the wave w_n^± propagates along the (Ox) axis from ∓∞ to ±∞. On the other hand, for n> N, w_n^± is exponentially  growing at ∓∞ and exponentially decaying at ±∞. For n=0,…,N, we consider the scattering of the wave w_n^+ by the obstacle located in . It is known that Problem (<ref>) admits a solution u_n=w_n^++u^s_n with the outgoing scattered field u^s_n written as

    u^s_n=∑_p=0^+∞ s^±_npw_p^±    ± x≥ L

with (s^±_np)∈^. The solution u_n is uniquely defined if and only if Trapped Modes (TMs) do not exist at the wavenumber k. We remind the reader that trapped modes are non zero  functions u∈^2(Ω) satisfying (<ref>), where ^2(Ω) is the space of square-integrable functions in Ω. We denote by 𝒦_t the set of k^2 such that TMs exist at the wavenumber k. On the other hand, the scattering coefficients s^±_np in (<ref>) are always uniquely defined, including for k^2∈𝒦_t. In the following, we will be particularly interested in the features of the reflection matrix  (whose size, determined by the number of propagative modes, depends on k)

    R(k):=(s^-_np)_0≤ n,p≤ N∈^N+1× N+1.
 
 Let  k∈(0;+∞)∖π. 
We say that the wavenumber k is reflectionless if R(k){0}.
 
Let us explain this definition. In general, by linearity, for an incident field 

    u_i=∑_n=0^Na_n w_n^+,     (a_n)_n=0^N∈^N+1,

Problem (<ref>) admits a solution u such that u=u_i+u_s with

    u_s=∑_p=0^+∞b_p^± w_p^±± x≥ L   b_p^±=∑_n=0^Na_ns^±_np∈.

The above definition says that, if k is reflectionless, then there are (a_n)_n=0^N∈^N+1∖{0} such that the b_p^- in (<ref>) satisfy b_p^-=0, p=0,…,N. In other words, the scattered field is exponentially decaying for x≤ -L. Let us notice finally that the corresponding total field u=u_i+u_s decomposes as

    [ u=∑_n=0^Na_n w_n^++ũ                x≤ -L; u=∑_n=0^Nt_n w_n^++ũ                 x≥ L;                      ]

where t_n=a_n+b_n^+ and where ũ decays exponentially for ± x≥ L. In other words, the total field is ingoing for x≤-L and outgoing for x≥ L.

In the following,  we call Reflectionless Modes (RMs) the functions u admitting expansion (<ref>) and we denote by 𝒦_r the set of k^2 such that the wavenumber k is reflectionless. The main objective of this article is to explain how to determine directly the set 𝒦_r and the corresponding RMs by solving an eigenvalue problem, instead of computing the reflection matrix for all values of k.  



§ CLASSICAL COMPLEX SCALING

As a first step, in this section we remind briefly how to use a complex scaling to compute trapped modes. Define the unbounded operator A of ^2() such that

    Au=-1γ Δ u

with Neumann boundary conditions ∂_yu=0y=0y=1. It is known that A is a selfadjoint operator (^2() is endowed with the inner product (γ ·,·)_^2()) whose spectrum σ(A) coincides with [0;+∞). More precisely, we have σ_ess(A)=[0;+∞) where σ_ess(A) denotes the essential spectrum of A. By definition, σ_ess(A) corresponds to the set of λ∈ for which there exists a so-called singular sequence (u^(m)), that is an orthonormal sequence (u^(m))∈^2(Ω)^ such that ((A-λ)u^(m)) converges to 0 in ^2(Ω). 
Besides, σ(A) may contain eigenvalues (at most a sequence accumulating at +∞) corresponding to TMs. In order to reveal these eigenvalues which are embedded in σ_ess(A), one can use a complex change of variables. For 0<θ<π/2, set η=e^iθ and define the function ℐ_θ:→ such that 

    ℐ_θ(x)=[  -L+(x+L) η       x≤- L;           x      |x|< L; + L+(x-L) η       x≥ L. ]

For the sake of simplicity, we  will use abusively the same notation ℐ_θ for the following map:
{→×(0;1),  
(x,y) ↦ (ℐ_θ(x),y)}.
Note that with this definition, the left inverse ℐ_θ^-1 of ℐ_θ, acting from ℐ_θ() to , is equal to  ℐ_-θ. One can easily check that for all n≥ 0, 
w_n^+∘ℐ_θ is exponentially decaying for x≥ L, while w_n^-∘ℐ_θ is exponentially decaying for x≤-L. As a consequence, defining from expansion (<ref>)  the function v_θ=u_s∘ℐ_θ, one has v_θ=u_s for |x|< L and v_θ∈^2(Ω) (which is in general not true for u_s). Moreover v_θ satisfies the following equation in :

    α_θ∂/∂ x(α_θ∂ v_θ/∂ x)+∂^2 v_θ/∂ y^2+k^2γ v_θ=k^2(1-γ)u_i

with α_θ(x)=1 for |x|<L and α_θ(x)=η^-1=η for ± x≥ L. In particular, for a TM, v_θ solves (<ref>) with u_i=0. This leads us to consider the unbounded operator A_θ of ^2() such that

    A_θv_θ=-1γ(α_θ∂/∂ x(α_θ∂ v_θ/∂ x)+∂^2 v_θ/∂ y^2)

again with homogeneous Neumann boundary conditions. Since α_θ is complex valued, the operator A_θ is no longer selfadjoint. However, we use the same definition as above for σ_ess(A_θ), which is licit for this operator. We recall below the main spectral properties of A_θ <cit.>:

i) There holds

    σ_ess(A_θ)=⋃_n∈, t≥0{n^2π^2+te^-2iθ}.

ii)  The spectrum of A_θ satisfies σ(A_θ)⊂ℛ_θ^- with 

    ℛ_θ^-:={z∈ | -2θ≤arg(z)≤ 0}.

iii)  σ (A_θ)∖σ_ess(A_θ) is discrete and contains only eigenvalues of finite multiplicity.

iv) Assume that k^2∈σ(A_θ)∖σ_ess(A_θ). Then k^2 is real if and only if k^2∈𝒦_t. Moreover if v_θ is an eigenfunction associated to k^2 such that m k^2<0, then v_θ∘ℐ_-θ is a solution of the original problem (<ref>) whose amplitude is exponentially growing at +∞ or at -∞.
 
The interesting point is that now TMs correspond to isolated eigenvalues of A_θ, and as such, they can be computed numerically as illustrated below. Note that the elements k^2 of σ(A_θ)∖σ_ess(A_θ) such that m k^2<0, if they exist, correspond to complex resonances (quasi normal modes). Let us point out that the complex scaling is just a technique to reveal them. Indeed complex resonances are intrinsic objects defined as the poles of the meromorphic extension from {z∈ |  m z>0} to {z∈ |  m z≤0} of the operator valued map z↦ (Δ +zγ)^-1. For more details, we refer the reader to <cit.>.





§ CONJUGATED COMPLEX SCALING


Now, we show that replacing the classical complex scaling by an unusual conjugated complex scaling, and proceeding  as in the previous section, we can define a new complex spectrum which contains the reflectionless values  k^2∈𝒦_r we are interested in. We define the map 𝒥_θ:→×(0;1) using the following  complex change of variables  

    𝒥_θ(x)=[  -L+(x+L) η       x≤- L;           x      |x|< L; + L+(x-L) η       x≥ L, ]

with again η=e^iθ (0<θ<π/2). Note the important difference in the definitions of 𝒥_θ and ℐ_θ for x≤-L: η has been replaced by the conjugated parameter η to select the ingoing modes instead of the outgoing ones in accordance with (<ref>). Now, if u is a RM associated to  k^2∈𝒦_r, setting  w_θ=u∘𝒥_θ, one has w_θ=u for |x|< L and w_θ∈^2(Ω) (which is not the case for u). The function w_θ satisfies the following equation in :

    β_θ∂/∂ x(β_θ∂ w_θ/∂ x)+∂^2 w_θ/∂ y^2+k^2γ w_θ=0

with β_θ(x)=1 for |x|< L, β_θ(x)=η for x≤-L and β_θ(x)=η for x≥ L. This leads us to define the unbounded operator B_θ of ^2() such that

    B_θw_θ=-1γ(β_θ∂/∂ x(β_θ∂ w_θ/∂ x)+∂^2 w_θ/∂ y^2)

with homogeneous Neumann boundary conditions. As A_θ, the operator B_θ is not selfadjoint. Its spectral properties are summarized in the following theorem which is proved in the last section of this article.
 
i) There holds 

    σ_ess(B_θ)=⋃_n∈, t≥0{n^2π^2+te^-2iθ, n^2π^2+te^+2iθ}.

ii) The spectrum of B_θ satisfies σ(B_θ)⊂ℛ_θ with 

    ℛ_θ:={z∈ | -2θ≤arg(z)≤ 2θ}.

iii) Assume that k^2∈σ(B_θ)∖σ_ess(B_θ). Then k^2 is real if and only if k^2∈𝒦_t∪𝒦_r. Moreover if w_θ is an eigenfunction associated to k^2 such that ± m k^2<0, then w_θ∘𝒥_-θ is a solution of (<ref>) whose amplitude is exponentially growing at ±∞ and exponentially decaying at ∓∞.
 
The important result is that isolated real eigenvalues of B_θ correspond precisely to TMs and RMs. The following proposition provides a criterion to determine whether an eigenfunction associated to a real eigenvalue of B_θ is a TM or a RM. 

Assume that (k^2,w_θ)∈×^2() is an eigenpair of B_θ such that k∈(Nπ;(N+1)π), N∈. Set

    (w_θ)=∑_n=0^N|∫_0^1w_θ(-L,y)φ_n(y) dy|^2

where φ_n is defined in (<ref>). If (w_θ)=0 then w_θ∘𝒥_-θ is a TM (k^2∈𝒦_t). If (w_θ)>0 then w_θ∘𝒥_-θ is a RM (k^2∈𝒦_r). In this case, the incident field u_i defined in (<ref>) with

    a_n=∫_0^1w_θ(-L,y)φ_n(y) dy,   n=0,…,N,

yields a scattered field which decays exponentially for x≤ -L.

The next proposition tells  that  B_θ satisfies the celebrated 𝒫𝒯 symmetry property when the obstacle is symmetric with respect to the (Oy) axis. This ensures in particular the stability of simple real eigenvalues, with respect to perturbations of the obstacle satisfying the same symmetry constraint. 

Assume that γ satisfies γ(x,y)=γ(-x,y) for all (x,y)∈. Then the operator B_θ is 𝒫𝒯-symmetric (𝒫𝒯B_θ𝒫𝒯=B_θ) with 𝒫φ(x,y)=φ(-x,y), 𝒯φ(x,y)=φ(x,y) for φ∈^2(). Therefore, we have σ(B_θ)=σ(B_θ).

The proof is straightforward observing that the β_θ defined after (<ref>) satisfies β_θ(-x,y)=β_θ(x,y).

Finally let us mention a specific difficulty which appears in the spectral analysis of B_θ. While Theorem <ref> guarantees that σ(A_θ)∖σ_ess(A_θ) is discrete, we do not write such a statement for the operator B_θ in Theorem <ref>. A major difference between both operators is that ∖σ_ess(A_θ) is connected whereas ∖σ_ess(B_θ) has a countably infinite number of connected components. As a consequence, to prove that σ (B_θ)∖σ_ess(B_θ) is discrete using the Fredholm analytic theorem, it is necessary to find one λ such that B_θ-λ is invertible in each of the components of ∖σ_ess(B_θ). In general, in presence of an obstacle, such a λ probably exists (proofs for certain classes of γ can be obtained working as in <cit.>). But for this problem, we can have surprising perturbation results. Thus, if there is no obstacle (γ≡1 in ), then there holds σ(B_θ)=ℛ_θ (see (<ref>)): all connected components of ∖σ_ess(B_θ), except the one containing the  complex half-plane e λ<0, are filled with eigenvalues. 
 To show this result, observe that for k^2∈σ(B_θ)∖σ_ess(B_θ), the function u∘𝒥_θ, with u(x,y)=e^ik x, is a non-zero element of B_θ. Notice that this pathological property is also true when  contains a family of sound hard cracks (homogeneous Neumann boundary condition) parallel to the (Ox) axis.




§ NUMERICAL EXPERIMENTS




 §.§  Classical complex scaling: classical complex resonance modes
 
We first compute the spectrum of the operator A_θ defined in (<ref>) with a classical complex scaling (complex resonance spectrum).  For the numerical experiments, we truncate the computational domain at some distance of the obstacle and use finite elements. This corresponds to the so-called Perfectly Matched Layers (PMLs) method. We refer the reader to <cit.> for the numerical analysis of the error due to truncation of the waveguide and discretization. The setting is as follows. We take γ such that γ=5 in 𝒪=(-1;1)×(0.25;0.75) and γ=1 in ∖𝒪 (see Figure <ref> (a)).  In the definition of the maps ℐ_θ, α_θ (see (<ref>), (<ref>)), we take θ=π/4 (so that η=e^iπ/4) and L=1. In practice, we use a P2 finite element method in the bounded domain _12={(x,y)∈ | -12<x<12} with Dirichlet boundary condition at x=±12. Finite element matrices are constructed with FreeFem++[FreeFem++, <http://www.freefem.org/ff++/>.]. 

In Figure <ref> and in the rest of the paper, we display the square root of the spectrum (k instead of k^2). The vertical marks on the real axis correspond to the thresholds (0, π, 2π, ...). In accordance with Theorem <ref>, we observe that √(σ(A_θ)) is located in the region √(ℛ_θ^-)={z∈ | -θ≤arg(z)≤ 0}. Moreover, the discretisation of the essential spectrum σ_ess(A_θ) defined in (<ref>) and forming branches starting at the threshold points appears clearly. Note that a simple calculation shows that √({n^2π^2+te^-2iθ, t≥0}) is a half-line for n=0 and a piece of hyperbola for n≥1. This is precisely what we get. 
Eigenvalues located on the real axis correspond to trapped modes (k^2∈𝒦_t).
 In the chosen setting, which is symmetric with respect to the axis ×{0.5}, one can prove that trapped modes exist <cit.>. 
On the other hand, the eigenvalues in the complex plane which are not the discretisation of the essential spectrum correspond to complex resonances. 





 §.§ Conjugated complex scaling: reflectionless modes
 
Now we compute the spectrum of the operator B_θ defined in (<ref>) with a conjugated complex scaling. First, we use exactly the same symmetric setting (see Figure <ref> (a)) as in the previous paragraph. In Figure <ref>, we display the square root of the spectrum √(σ(B_θ)). Since γ satisfies γ(x,y)=γ(-x,y), according to Proposition <ref> we know that B_θ is 𝒫𝒯-symmetric and that therefore its spectrum is stable by conjugation (σ(B_θ)=σ(B_θ)). This is indeed what we obtain. Note that the mesh has been constructed so that 𝒫𝒯-symmetry is preserved at the discrete level. 𝒫𝒯-symmetry is an interesting property in our case because it guarantees that eigenvalues located close to the real axis which are isolated (no other eigenvalue in a vicinity) are real. Therefore, according to Theorem <ref>, they correspond to trapped modes or to reflectionless modes. 
Remark that, for the same geometry, the spectrum of B_θ (Figure <ref>) contains more elements on the real axis than the spectrum of A_θ (Figure <ref>): the additional elements (green points in Figure <ref>) correspond to reflectionless modes. 


In Figure <ref> top, we represent the real part of eigenfunctions associated with seven real eigenvalues of B_θ. 
To obtain these pictures, we take L=4 in the definition of 𝒥_θ in (<ref>)
and we display only the restrictions of the eigenfunctions to _L={(x,y)∈ | -L<x<L}. 
We recognize two trapped modes (images 3 and 5). 
The other modes are reflectionless modes. In Figure <ref> bottom, we provide the value of the indicator function  defined in (<ref>) for the seven eigenmodes. We have to mention that eigenmodes are normalized so that their ^2 norm  is equal to one. The indicator function  offers a clear criterion to distinguish between trapped modes and reflectionless modes. 
Moreover, in order to inspect the scattering coefficient, we remark that for reflectionless modes associated with wavenumbers k smaller than π, the incident field u_i in (<ref>) decomposes only on the piston mode w^+_0(x,y)=e^ikx/√(2k) (monomode regime). In this case the reflection matrix R(k) in (<ref>) is nothing but the usual reflection coefficient. 
In Figure <ref>, we thus display the modulus of this coefficient R_00(k) with respect to k∈(0.1;3.1). As expected, we observe that R_00 vanishes for the values of k obtained in Figure <ref> solving the spectral problem for B_θ. 
Of course obtaining the curve k↦ |R_00(k)| is relatively costly and it is precisely 
what we want to avoid by computing the reflectionless k as eigenvalues. Here it is simply a way to check our results. 

In Figure <ref>, we represent the modulus of reflectionless mode eigenfunctions of B_θ associated with one real eigenvalue and two complex conjugated eigenvalues. We observe, and this is true in general, a symmetry with respect to the (Oy) axis for modes corresponding to real eigenvalues which disappears for complex ones. This is the so-called broken symmetry phenomenon which is well-known for 𝒫𝒯-symmetric operators (see e.g. the review <cit.>).







Now, we use the  non-symmetric setting (see Figure <ref> (b)), 
and we display the square root of the spectrum of B_θ (in Figure <ref>)
for a coefficient γ which is not symmetric in x nor in y. 
More precisely, we take γ such that γ=5 in 𝒪=(-1;0]×(0.25;0.5)∪[0;1)×(0.25;0.75) and γ=1 in ∖𝒪. We observe that the  spectrum is no longer stable by conjugation (σ (B_θ)σ (B_θ)) since
the operator B_θ is not  𝒫𝒯-symmetric, and there is no “help” for the eigenvalues to be real.
However, a closer look shows the presence of eigenvalues close to the real axis,
in particular for k≈1.0 + 0.13i, k≈1.9 + 0.005i, k≈2.5 + 0.02i, k≈2.8 + 0.08i and k≈3.0 - 0.008i. In Figure <ref>, we represent k↦|R_00(k)| for k∈(0.1;3.1) where there is only one propagating mode in the leads.
 It is interesting to note that the above computed complex reflectionless modes (located close to the real axis)   have an influence on this curve. More precisely, k↦|R_00(k)| attains minima for k∈(0.1;3.1) close to the real part of these complex reflectionless modes. Therefore complex reflectionless modes also have  significance for scattering at real frequncies.









§ PROOFS

Finally we give the proofs of Theorem <ref> and Proposition <ref>.

Proof of Theorem <ref>. 
i) First, let us explain how to show that for t≥0 and n∈, n^2π^2+te^-2iθ belongs to σ_ess(B_θ). 
Consider a smooth cut-off function χ defined on  such that χ(x)=0 for |x|>L and χ_^2()=1. For m≥1, set v_+^(m)(x,y)=m^-1/2χ((x-m^2L)/m)e^i√(t)xφ_n(y) where φ_n is defined in (<ref>). One can check that (v_+^(m)) is a singular sequence for B_θ at n^2π^2+te^-2iθ (note that the supports of v_+^(m) and v_+^(m') for m≠ m' do not overlap). On the other hand, we prove that (v_-^(m)), with v_-^(m)(x,y)=v_+^(m)(-x,y), is a singular sequence
for B_θ at n^2π^2+te^+2iθ. Summing up, we obtain
∪_n∈, t≥0{n^2π^2+te^-2iθ, n^2π^2+te^+2iθ}⊂σ_ess(B_θ).

The converse inclusion requires a bit more work. Observe that if 
λ∈σ_ess(B_θ), then one can prove by localization the existence of a corresponding singular sequence (v^(m)) supported either in x>L or in x<-L. In the first case, it means that (v^(m)) is also a singular sequence for A_θ, so that λ∈σ_ess(A_θ). In the second case, (v^(m))  is a singular sequence for the complex conjugate of A_θ, which implies that λ∈σ_ess(A_θ). Finally, the result follows from item i) of Theorem <ref>. 


ii) Define the form b_θ(·,·)   such that 

    b_θ(u,v)=∫_β_θ∂_xu ∂_xv+β^-1_θ∂_yu ∂_yv-k^2 β^-1_θu v dxdy.

One can check that for k^2∈∖ℛ_θ, the numbers 1, e^± iθ, -k^2, -k^2 e^± iθ are all located in some region {z∈ |  e (e^iκz)≥μ} for some κ∈ [0;2π) and μ>0. We deduce that 

    e (e^iκb_θ(u,u))≥μ∫_|u|^2+|∂_xu|^2+|∂_yu|^2 dxdy

which shows that b_θ(·,·) is coercive on the Sobolev space ^1(Ω). As a consequence of the Lax-Milgram theorem, B_θ-k^2 is invertible for k^2∈∖ℛ_θ, which guarantees that σ(B_θ)⊂ℛ_θ.

iii) If k^2∈𝒦_t∪𝒦_r, then  k^2 is real and by construction k^2∈σ(B_θ)∖σ_ess(B_θ). Conversely, assume that k^2 is real and that k^2∈σ(B_θ)∖σ_ess(B_θ). There is one N∈ such that k∈(Nπ;(N+1)π). Consider a non zero  w_θ∈ (B_θ-k^2). Setting u=w_θ∘𝒥_-θ, we find that u satisfies Δ u+k^2γ u=0 in  and expands as

    u=[ ∑_n=0^N a^-_n w_n^++∑_n=N+1^+∞ a^-_n w_n^-   x≤ -L;                       ∑_n=0^+∞ a^+_n w_n^+   x≥ L, ]

with (a^±_n)∈^. If one of the a^-_n, for n=0,…,N, is non zero, it means that u is a RM. If on the contrary a^-_n=0 for all n=0,…,N, then one can prove that a^+_n=0 for all n=0,…,N, so that u is a TM. Indeed, integrating by parts, we show that the quantity

    ℱ=∫_Σ_-L∪Σ_L (∂_x u u-u ∂_x u)  dy,

with Σ_± L={± L}×(0;1), satisfies ℱ=0. On the other hand, a direct calculation using expansion (<ref>) and the orthonormality of the φ_n yields 
ℱ=∑_n=0^N i(|a_n^+|^2+|a_n^-|^2)
and the result follows.


Finally, consider some k^2∈σ(B_θ)∖σ_ess(B_θ) such that m k^2>0 (the case m k^2<0 is similar). There is a unique N∈ such that 2π-2θ<arg(k^2-(Nπ)^2)<2π and π<arg(k^2-((N+1)π)^2)<2π-2θ. Then if w_θ∈ (B_θ-k^2)∖{0}, expansion (<ref>) for u=w_θ∘𝒥_-θ holds and one of the a^-_n, n=0,…,N, has to be non zero. Indeed, otherwise u would be exponentially decaying for ± x≥ L and k^2 would be in σ(A), which is impossible because σ(A)=[0;+∞). Therefore the amplitude of u is exponentially decaying at +∞ and exponentially growing at -∞.□

Proof of Proposition <ref>. 
If (k^2,w_θ)∈×^2(Ω) is an eigenpair of B_θ, then  u=w_θ∘𝒥_-θ expands as in (<ref>). Moreover, we deduce from the orthogonality of the φ_n that 

    a_n^-=∫_0^1w_θ(-L,y)φ_n(y) dy,   n=0,…,N,

which gives the result, using the same arguments as in the proof of Theorem <ref>, item iii). 
□



§ CONCLUDING REMARKS

It is often desirable to determine frequencies for which a wave can be completely transmitted through a structure, a task usually leading to the tedious work of evaluating the scattering coefficients for each frequency. Here, we have shown that reflectionless frequencies can be directly computed as the eigenvalues of a non-selfadjoint operator B_θ (see (<ref>)) with conjugated complex scalings enforcing ingoing behaviour in the incident lead and outgoing behaviour in the other lead. The reflectionless spectrum of this operator B_θ provides a  complementary information to the one contained in the classical complex resonance spectrum associated with leaky modes which decompose only on outgoing waves (see the operator A_θ in (<ref>)). Note that eigenvalues corresponding to trapped modes belong to both the reflectionless spectrum and to the classical complex resonance spectrum because trapped modes do not excite propagating waves.

Moreover, since ingoing and outgoing complex scalings can be associated, respectively, with gain and loss, one observes that the non-selfadjoint operator B_θ leads to consider a natural 𝒫𝒯 symmetric problem when the structure has mirror symmetry. Interestingly, a direct calculus shows that in the very simple case of a 1D transmission problem through a slab of constant index, reflectionless frequencies are all real. This gives an example of a non-selfadjoint 𝒫𝒯 symmetric operator with only real eigenvalues.

In this work, we investigated scattering problems in waveguides with N=2 leads for which two reflectionless spectra exist: one associated with incident waves propagating from the left and another corresponding to incident waves propagating from the right. The more general case with N (N ≥ 2) leads can be considered as well. Among the total of 2^N different spectra with an ingoing or an outgoing complex scaling in each lead, two spectra correspond to eigenmodes which decompose on waves which are all outgoing or all ingoing. As a consequence, there are 2^N-2 reflectionless spectra.





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