Resonances in a Dirichlet quantum waveguide coupled to a cavity

Resonances in a Diric hlet quan tum w a v eguide coupled to a ca vit y Sylwia Kondej 1 ∗ , Nik oloz Kurtskhalia 2 1 Institute of Physics, Univ ersit y of Zielona G´ ora, ul. Szafrana 4a, 65246 Zielona G´ ora, P oland, e-mail: s.kondej@if.uz.zgor a.pl 2 Sc ho ol of Ph ysics, F ree Univ ersity of Tbilisi, 240 Davit Aghmashenebeli Alley , Tbilisi 0159, Georgia, e-mail: nkurt22@fr e euni.e du.ge De dic ate d to Pr ofessor Pavel Exner on the o c c asion of his 80th birthday. Abstract W e consider a Diric hlet wa veguide in R n ( n = 2 , 3) with an attached cavit y . W e sho w that if the cavit y admits a small gap, then the original embedded eigenv alues turn into resonances. The main question w e address is how the size of the gap affects the resonant prop erties, in particular the imaginary part of the resonan t p ole. F or example, in the case of a t wo dimensional wa veguide with a gap of size ε , w e show that the leading order term of the resonance b eha v es as O ( ε 2 ). In the three-dimensional case, if the ap erture is defined by a rectangular op ening with volume prop ortional to ε 2 , the resonant comp onen t b eha ves as O ( ε 4 ). This shows that, in the analyzed class of mo dels, the characteristic time scale asso ciated with the resonances is generically of order O ((v ol ε ) − 2 ), where vol ε denotes the volume of the ap erture inducing the resonance. Keyw ords: Hamiltonian of quantum system, Diric hlet wa v eguides, embedded eigen v alues, resonances. Mathematics Sub ject Classification: 47B38, 81Q10, 81Q15, 81Q80 1 In tro duction In this paper, w e consider a class of mo dels inv olving semi-infinite straigh t w av eguides in R n , with n = 2 , 3, equipp ed with a resonan t cavit y . T o ensure clarity of presen tation, we begin by describing a tw o-dimensional planar semi-infinite wa veguide Σ of width d 2 : Σ = { ( x 1 , x 2 ) : x 1 ∈ [0 , ∞ ) , x 2 ∈ [ 0 , d 2 ] } , b earing in mind that the results obtained in this pap er are applicable to a more general situation, which will b e described later. The wa v eguide is equipped with a cavit y of width d 1 , situated at the closed end, see Fig. 1. The righ t-hand w all of the cavit y con tains a gap of size ε > 0; more precisely , it is defined as I ε := { ( d 1 , x 2 ) : x 2 ∈ [ 0 , t ] ∪ [ t + ε , d 2 ] } , where t ≥ 0, and t + ε ≤ d 2 . In particular, I 0 = { ( d 1 , x 2 ) : x 2 ∈ [ 0 , d 2 ] } and the gap is determined by ¯ I ε = I 0 \ I ε . The Hamiltonian of the system is defined as the Dirichlet Laplacian − ∆ D I ε acting in L 2 (Σ) and sub ject to Dirichlet b oundary conditions on ∂ Σ ∪ I ε . ∗ Corresponding author. 1 Figure 1: Geometry of the wa veguide Σ with a cavit y C containing a gap ¯ I ε . The threshold of the con tinuous spectrum is determined by the lo west transv erse mo de; therefore, the essen tial sp ectrum takes the form σ ess ( − ∆ D I ε ) = h π d 2  2 , ∞  . (1.1) F or a closed ca vity , i.e., when ε = 0, the trapped mo des give rise to discrete energy levels ξ l,k =  π l d 1  2 +  π k d 2  2 , whic h, in fact, represent eigenv alues embedded in the essential sp ectrum. In tro ducing a small ap erture in the right hand hard wall of the cavit y enables the p ossibilit y of quantum tunnelling from the cavit y into the infinite c hannel of the wa veguide. As a result, the previously trapped mo des become metastable. It is natural to exp ect that the size of the gap is related to the characteristic time scale asso ciated with these metastable states. The problem can b e form ulated in terms of resonances, where the width of a resonance is inv ersely prop ortional to the corresp onding the characteristic time scale. The main question addressed in this paper is the following: What is the asymptotic b ehavior of the r esonanc e width as a function of ε ? A t this stage, we would like to emphasize that in the current pap er, w e also tackle the problem of a three-dimensional wa veguide with the geometry illustrated in Fig. 2. In this case, the gap ¯ I ε is defined b y a rectangle with dimensions ε and aε , where a > 0 is a constant. Figure 2: Geometry of the wa v eguide Σ with a b o x ca vity C containing a rectangular gap ¯ I ε . 2 T o address the problem of resonances, w e formulate the sp ectral problem in terms of the kernel of a certain operator K ε ( z ), which characterizes the presence of an eigenv alue. Specifically , if k er K ε ( z 0 )  = ∅ , then z 0 is an eigen v alue of the Dirichlet Laplacian − ∆ D I ε . This criterion can be considered as a realization of the Birman–Sc hwinger principle in systems with a v arying spatial structure, in particular those inv olving Diric hlet b oundary conditions. Motiv ated by this characterization, we in vestigate whether there exist com- plex v alues z for which k er K ε ( z )  = ∅ , in terpreted as resonance p oles in the complex plane. The main results of the pap er can b e formulated as follo ws. Main Results. Assume that in a tw o-dimensional planar wa veguide, an embedded eigenv alue ξ l,k , given by (1.1), has m ultiplicit y N. Then, under a small perturbation induced b y the opening of the gap, there exist N complex-v alued functions ε 7→ z j ( ε ), for j = 1 , . . . , N, suc h that k er K ε  z j ( ε )   = ∅ , and eac h z j ( ε ) admits the asymptotic expansion z j ( ε ) = ξ l,k + µ j ( ε ) + i ν j ( ε ) , where the real-v alued functions µ j ( ε ) and ν j ( ε ) satisfy µ j ( ε ) = O ( ε 2 ) , ν j ( ε ) = O ( ε 2 ) as ε → 0 . These complex quan tities z j ( ε ) determine the resonance poles, with the imaginary parts describing the deca y rates of the corresp onding metastable states. In three-dimensional case defined in the previous discussion the result holds true with µ j ( ε ) = O ( ε 4 ) and ν j ( ε ) = O ( ε 4 ). In our view, the imp ortance of the present problem stems from tw o main reasons. First, resonances pla y a fundamen tal role in determining the characteristic time scale of metastable states, which in turn influence quantum transp ort prop erties. The results presented in this pap er allow us to conclude that, for the considered types of w av eguides, the characteristic time scale behav es as τ = O ( | ¯ I ε | − 2 ) , where | ¯ I ε | denotes the volume of the ap erture region. A precise understanding of resonances features enables con trol ov er the dynamical b eha vior of quantum particles in w av eguides with the structure of Σ. This, p oten tially , op ens the w ay to optimization of quantum and electronic devices where transp ort characteristics can be tuned through geometric mo difications. A second motiv ation for this study arises from a mathematical p ersp ectiv e. T o rigorously description the resonance phenomena, it is necessary to dev elop a set of sp ectral to ols in this setting. The main analytical c hallenge comes from the fact that the geometry of the domain v aries with the size of the segment I ε in the w all. This do es not allow a direct application of tec hniques work ed out in so-called soft w a veguide models, cf. [19, 22]. T o ov ercome this difficult y , w e develop several auxiliary results using p erturbation theory , a generalized version of the Implicit F unction Theorem, analytic contin uation metho ds for op erators, and asymptotic techniques that allo w for the analysis of the asymptotic b eha vior of resonance poles. W e b eliev e that the mathematical framew ork developed here and applied to quan tum systems may b e of interest to b oth the mathematical and physical comm unities, as it provides insigh t in to the relation b et ween geometry , sp ectral theory , and resonance phenomena. T o conclude, let us briefly commen t on the curren t state of the art concerning the similar problems. There exists a large n umber of papers on Diric hlet wa veguides; see, for example, [11] and references therein. The sp ectral structure of v arious t yp es of w av eguides has b een thoroughly in v estigated, particularly in relation to how the geometry of a wa veguide influences the existence of discrete spectrum — see, e.g., [8, 10, 11]. Metho ds for analyzing wa veguides with spatially v arying geometry ha ve b een developed in [30]. 3 Em b edded eigen v alues and resonances hav e b een studied from v arious p ersp ectiv es in w av eguide-like structures and in configuration spaces with in teractions lo calized on low er-dimensional subsets; see, for instance, [6, 16, 20, 23, 24, 27, 28]. Resonant tunneling effects in wa veguides weakly coupled through small geometric op enings were inv estigated, among others, in [3], where resonances generated by distant Dirichlet barriers were analyzed, as well as in [5, 7], which studied almost complete transmission through perforated screens or barriers with small apertures. In particular, Delitsyn and Grebenko v [7] demonstrated ho w specific geometric configurations lead to resonan t enhancement of reaction rates, revealing close connections betw een diffusion dynamics, sp ectral prop erties of the asso ciated op erators, and near-complete transmission effects. The pap er is organized as follo ws. In Section 2, w e formulate the problem in a tw o-dimensional setting and obtain preliminary sp ectral results. In Section 3, w e analyze and discuss the existence of resonances. Section 4 is dev oted to the asymptotic analysis of the p erturbativ e components of the resonance p oles. In Section 5, w e extend the problem to a three-dimensional setting and presen t the main results. Finally , Section 6 contains conclusions, final remarks, and op en questions. 2 Description of mo del and preliminary results in t w o-dimensional system 2.1 Geometry of planner w av eguide and Hamiltonian of system W e consider a one-sided infinite, planar wa v eguide with a width of d 2 : Σ := { ( x 1 , x 2 ) : x 1 ∈ [0 , ∞ ) , x 2 ∈ [ 0 , d 2 ] } . F urthermore, we equip Σ with a partition I ε ⊂ Σ, with a aperture of the size ε ≥ 0, defined by I ε := { ( d 1 , x 2 ) : x 2 ∈ [ 0 , t ] ∪ [ t + ε , d 2 ] } , where d 1 > 0, t ≥ 0 and t + ε ≤ d 2 . F or a special case ε = 0 we hav e I 0 = { ( d 1 , x 2 ) : x 2 ∈ [ 0 , d 2 ] } . In the following we will use the notation C = { ( x 1 , x 2 ) : x 1 ∈ [ 0 , d 1 ] , x 2 ∈ [ 0 , d 2 ] } . The Hamiltonian in the w av eguide system is defined as − ∆ D I ε : W 2 , 2 0 (Σ \ I ε ) → L 2 (Σ) , where W 2 , 2 0 (Σ \ I ε ) denote the Sob olev space with Dirichlet b oundary conditions on ∂ Σ ∪ I ε . 2.2 Preliminary results for the sp ecial cases W e consider tw o sp ecial cases: first ε = d 2 , with t = 0 and second d 1 > 0, ε = 0. In the first case, w e deal with the wa veguide without a cavit y , while in the second one, the cavit y is determined by an en tirely closed hard w all lo cated at the p oin t ( d 1 , x 2 ), x 2 ∈ [0 , d 2 ]. The Hamiltonian corresp onding to the situation ε = d 2 is giv en by − ∆ D ≡ − ∆ D I d 2 : W 2 , 2 0 (Σ) → L 2 (Σ). Its resolv ent R ( z ) := ( − ∆ D − z ) − 1 is an integral op erator with the k ernel (Green’s function) given by G ( z ; ( x 1 , x 2 ) , ( y 1 , y 2 )) := ∞ X n =1 G n ( z ; x 1 , y 1 ) e 0 n ( x 2 ) e 0 n ( y 2 ) , (2.1) where G n ( z ; x 1 , y 1 ) = e i √ z − ( πn/d 2 ) 2 | x 1 + y 1 | i p z − ( π n/d 2 ) 2 − e i √ z − ( πn/d 2 ) 2 | x 1 − y 1 | i p z − ( π n/d 2 ) 2 , ℑ p z − ( π n/d 2 ) 2 > 0 , (2.2) 4 and n ∈ N , with N defined as p ositiv e integers; (we keep the absolute v alue in the ab o ve form ula although x 1 + y 1 > 0), moreov er e 0 n ( · ) := r 2 d 2 sin  π n d 2 ·  . (2.3) W e define also the operator R ε ( z ) : L 2 (Σ) → L 2 (Σ) with the integral kernel G ε ( z ; ( x 1 , x 2 ) , ( y 1 , y 2 )) := ∞ X n =1 G n ( z ; x 1 , y 1 ) e ε n ( x 2 ) e ε n ( y 2 ) , (2.4) where e ε n ( · ) := e 0 n ( · ) χ I ε ( · ) = r 2 d 2 sin  π n d 2 ·  χ I ε ( · ) . (2.5) and χ I ε ( · ) stands for the c haracteristic function of I ε . T o form ulate the first results we define K ε ( z ) : L 2 ( I ε ) → L 2 ( I ε ) as an in tegral op erator with the k ernel and K ε ( z ; x 2 , y 2 ) giv en b y K ε ( z ; x 2 , y 2 ) := G ε ( z ; ( d 1 , x 2 ) , ( d 1 , y 2 )) = ∞ X n =1 G n ( z ; d 1 , d 1 ) e ε n ( x 2 ) e ε n ( y 2 ) , (2.6) where G n ( z ; d 1 , d 1 ) = e i 2 d 1 √ z − ( πn/d 2 ) 2 i p z − ( π n/d 2 ) 2 − 1 i p z − ( π n/d 2 ) 2 , cf. (2.2) and (2.4). In fact, op erator K ε ( z ) determines a bilateral embedding of R ε ( z ) to the space L 2 ( I ε ). Throughout the pap er, the op erator K ε ( z ) is defined as a b ounded op erator on L 2 ( I ε ) equipp ed with its natural top ology . Ho wev er, in the Birman–Sch winger argumen t b elo w, in order to ensure injectivity and sufficien t regularit y of the asso ciated single-la yer potential, we consider K ε ( z ) as acting on the space H 1 / 2 ( I ε ). Theorem 2.1 A numb er z 0 is an eigenvalue of − ∆ D I ε if and only if k er K ε ( z 0 )  = { 0 } , wher e the op er ator K ε ( z ) is c onsider e d as acting in H 1 / 2 ( I ε ) . Mor e over, dim ker  − ∆ D I ε − z 0  = dim ker K ε ( z 0 ) . (2.7) Pro of. Assume first that f ∈ ker K ε ( z 0 ) ⊂ H 1 / 2 ( I ε ). Define g ( x 1 , x 2 ) := R ( z 0 )   Σ ,I ε f = Z I ε G ε  z 0 ; ( x 1 , x 2 ) , ( d 1 , y 2 )  f ( y 2 ) d y 2 , (2.8) By the mapping prop erties of the single-la yer p oten tial (see [25, Thm. 4.18]), the function g b elongs to H 1 loc (Σ \ I ε ) and satisfies ( − ∆ D − z 0 ) g = 0 there. Interior elliptic regularit y (see [25, Thm. 2.27]) then yields 1 g ∈ H 2 (Σ \ I ε ). F or ( x 1 , x 2 ) ∈ I ε , that is x 1 = d 1 and x 2 ∈ [0 , t ] ∪ [ t + ε, d 2 ], w e ha ve g ( d 1 , x 2 ) = ( K ε ( z 0 ) f )( x 2 ) = 0 , 1 Although the mapping properties of the single-la yer p oten tial are usually formulated for densities supp orted on closed Lipschitz hypersurfaces, the same argument applies verbatim on every op en set U ⋐ Σ \ I ε , since dist( U, I ε ) > 0 and the Green kernel is smo oth in the field v ariable aw ay from the supp ort. 5 and clearly g = 0 on ∂ Σ. Therefore g ∈ D ( − ∆ D I ε ). Moreov er, ( − ∆ D I ε − z 0 ) g = 0 in Σ \ I ε . (2.9) Th us each f ∈ ker K ε ( z 0 ) giv es rise to an eigenfunction g ∈ ker( − ∆ D I ε − z 0 ). Con versely , let g ∈ D ( − ∆ D I ε ) satisfy (2.9). Then g = 0 on ∂ Σ ∪ I ε . By Green’s represen tation form ula for the Dirichlet Laplacian, there exists f ∈ H 1 / 2 ( I ε ) suc h that g = R ( z 0 )   Σ ,I ε f . Restricting g to I ε yields 0 = g ( d 1 , x 2 ) = ( K ε ( z 0 ) f )( x 2 ) , x 2 ∈ [0 , t ] ∪ [ t + ε, d 2 ] , and hence f ∈ ker K ε ( z 0 ). The ab o ve construction establishes a one-to-one corresp ondence b et ween ker K ε ( z 0 ) and ker( − ∆ D I ε − z 0 ), whic h prov es (2.7). Remark 2.2 Using the argument from the pro of of Theorem 2.1, one concludes that for any z b elonging to the resolv ent set ρ ( − ∆ D I ε ) and any function g ∈ D ( − ∆ D I ε ), there exists f ∈ W 2 , 2 (Σ) ∩ W 1 , 2 0 (Σ) suc h that g = f − R ( z )   Σ ,I ε K ε ( z ) − 1 I f . (2.10) Here I denotes the trace op erator I : W 2 , 2 (Σ) − → H 1 / 2 ( I ε ) , and the op erator K ε ( z ) is inv ertible since z ∈ ρ ( − ∆ D I ε ). Moreo ver, the representation (2.10) holds in W 2 , 2 (Σ \ I ε ). Note that Theorem 2.1 represen ts the limiting case of the Birman–Sc h winger principle for a regular or delta p oten tial, cf. [1, 2, 12, 13, 29]. In the whole-space case, analogous settings are describ ed by Kre ˘ ın-t yp e resolv ent form ulae, linking resolv ent poles to Birman–Sc hwinger conditions; see [26, Sec. 7.1]. F or example, if the delta barrier is defined b y a coupling constan t α > 0 and it is localized on I ε , then the Birman–Sc hwinger principle tak es the form k er  α − 1 + K ε ( z )   = ∅ . F ormally taking the limit α → ∞ , whic h mo dels Dirichlet b oundary conditions, we get the equation (2.7). 2.3 Em b edded eigen v alues In a sp ecial case, if ε = 0, i.e. the ca vity C := { ( x 1 , x 2 ) : x 1 ∈ [0 , d 1 ] , x 2 ∈ [0 , d 2 ] } is closed, then the Hamiltonian − ∆ D I 0 decouples − ∆ D I 0 = − ∆ D ( C ) ˙ +  − ∆ D (Σ \ C )  , where ∆ D ( C ) is the Diric hlet Laplacian in L 2 ( C ) and ∆ D (Σ \ C ) in L 2 (Σ \ C ). The op erator − ∆ D C has discrete sp ectrum given b y ξ l,k :=  π l d 1  2 +  π k d 2  2 , k , l ∈ N . (2.11) On the other hand, the essential spectrum of the Hamiltonian in wa veguide has a form of the half line and remains stable in the presence of the ca vity , i.e. σ ess ( − ∆ D ) = σ ess ( − ∆ D (Σ \ C )) = σ ess ( − ∆ D I 0 ) = σ ess ( − ∆ D I ε ) = h π 2 d 2 2 , ∞  . (2.12) 6 This means that all the num b ers (2.11) determine the embedded eigenv alues of − ∆ D I 0 . Note, that this result is consistent with Theorem 2.1. Indeed, assume that z 0 = ξ l,k and note e i 2 d 1 √ ξ l,k − ( π k/d 2 ) 2 = e i 2 π l = 1. Therefore G k ( z 0 = ξ l,k ; d 1 , d 1 ) = 0 . This implies e 0 k ∈ ker K 0 ( z 0 = ξ l,k ). The function g ( x 1 , x 2 ) = Z I 0 G ( z 0 = ξ l,k ; ( x 1 , x 2 ) , ( d 1 , y 2 )) e 0 k ( y 2 ) d y 2 ∈ L 2 ( C ) , can be expanded as follows g ( x 1 , x 2 ) = ∞ X n =1 Z I 0 G n ( z 0 = ξ l,k ; x 1 , d 1 ) e 0 n ( x 2 ) e 0 n ( y 2 ) e 0 k ( y 2 )d y 2 = G k ( z 0 = ξ l,k ; x 1 , d 1 ) e 0 k ( x 2 ) = d 1  e iπ l | x 1 + d 1 | /d 1 iπ l − e iπ l | x 1 − d 1 | /d 1 iπ l  e 0 k ( x 2 ) = d 1 iπ l  e iπ l ( x 1 + d 1 ) /d 1 − e − iπ l ( x 1 − d 1 ) /d 1  e 0 k ( x 2 ) = 2 d 1 π l e ilπ sin  π lx 1 d 1  e 0 k ( x 2 ) = d 3 / 2 1 π l √ 2 e iπ l e 0 l ( x 1 ) e 0 k ( x 2 ) , where e 0 l ( x 1 ) = q 2 d 1 sin  π lx 1 d 1  , i.e. we use the same notation for the longitudinal and transverse cavit y eigenstates, remembering that the former is asso ciated with d 1 , and the latter with d 2 . The ab o ve calculus sho ws that e 0 l ( · ) e 0 k ( · ) χ C \ Σ determine the eigenfunctions of − ∆ D I 0 corresp onding to ξ l,k =  π l d 1  2 +  π k d 2  2 , whic h are em b edded into essential sp ectrum, see (2.12). Remark 2.3 De gener acy of the emb e dde d eigenvalues. It is well known that the higher dimensional infinite w ell potential can admit degenerate eigenstates and, from the persp ectiv e of further discussion, it is important to address this issue here. The simplest example occurs when d 1 = d 2 . In this case, an y eigenv alue ξ l,k is at least doubly degenerate, except for the diagonal case l = k , corresp onding to the pair of eigenfunctions e 0 l ( x 1 ) e 0 k ( x 2 ) and e 0 k ( x 1 ) e 0 l ( x 2 ). Of course, higher degrees of degeneracy are also p ossible. F or example, if for a pair ( l , k ) there exists another pair ( l ′ , k ′ ) suc h that l 2 + k 2 = l ′ 2 + k ′ 2 , (2.13) then the corresponding energy level is degenerate. An example of suc h a case is ( l, k ) = (7 , 1) and ( l ′ , k ′ ) = (5 , 5), since 7 2 + 1 = 5 2 + 5 2 = 50 . Therefore, the energy level corresp onding to π 2 d 2 1 50 has a degeneracy of 3. In the case of the pair (1 , 18) there exist tw o more pairs that satisfy (2.13); namely , (6 , 17) and (10 , 15), i.e. the state corresp onding to π 2 d 2 1 325 has a degeneracy of 6. 3 Resonances caused b y an ap erture in ca vit y In fact, Theorem 2.1 shows that the problem of determining the eigen v alues of the operator − ∆ D I ε , i.e., the real poles of its resolv ent, can b e reformulated in terms of the kernel of K ε ( z ). In this section, w e take this 7 analysis one step further and ask whether the op erator K ε ( z ) admits an analytic contin uation to the second Riemann sheet, and whether there exists a p oin t z with ℑ z ≤ 0 suc h that k er K ε ( z )  = ∅ . The operator K ε ( z ), see (2.6), can written as K ε ( z ) = ∞ X n =1 G n ( z ; d 1 , d 1 ) P ε n : L 2 ( I ε ) → L 2 ( I ε ) , (3.1) where P ε n = ( e ε n , · ) L 2 ( I ε ) e ε n . Although K ε ( z ) acts on the space L 2 ( I ε ), ho wev er it can b e naturally extended to the space L 2 ( I 0 ) b y determining the functions { e ε k } ∞ k =1 , cf. (2.5), as elements of L 2 ( I 0 ). Of course, the functions e ε k do not form a basis in L 2 ( I 0 ). In the following discussion, we will also use v arious modifications of e 0 k . T aking the opportunity , b efore pro ceeding with the analysis, we would like to clarify some related notions: e ε k = e 0 k χ I ε , ˘ e ε k := e 0 k χ ¯ I ε , k ∈ N . (3.2) 3.1 Analytic con tin uation of K ε ( z ) Our aim is to in v estigate k er K ε ( z ) for v alues of z close to the original em b edded eigen v alue ξ l,k . F or this aim, w e introduce the notation B ( ξ l,k ) to denote a small neighborho od of ξ l,k , and w e assume that z ∈ B ( ξ l,k ), i.e., z = ξ l,k + δ for some δ ∈ C with | δ | sufficiently small. T o build the analytic contin uation, we distinguish t wo cases. Namely , for the comp onen ts in (3.1) lab eled by n suc h that  π n d 2  2 < ξ l,k =  π l d 1  2 +  π k d 2  2 , w e construct the analytic con tinuation of B ( ξ l,k ) ∋ z 7→ p z − ( π n/d 2 ) 2 to the low er second sheet through [( π n/d 2 ) 2 , ∞ ). Then, ℑ p z − ( π n/d 2 ) 2 < 0 when δ mo v es to the low er half-plane. On the other hand, for n such that ( π n/d 2 ) 2 > ξ l,k , the function B ( ξ l,k ) ∋ z 7→ p z − ( π n/d 2 ) 2 is analytic for both p ositiv e and negative ℑ δ and do es not require a sp ecial analytic contin uation. If ( π n/d 2 ) 2 = ξ l,k and ℜ δ > 0, then we pro ceed as in the first case, i.e. in this situation, ℑ p z − ( π n/d 2 ) 2 < 0, and if ℜ δ < 0, then the square root expression leav es on the first Riemann sheet, i.e. ℑ p z − ( π n/d 2 ) 2 > 0. Note that in this case, G n ( z ; d 1 , d 1 ) = O ( δ ), and if δ = 0, then G n ( z ; d 1 , d 1 ) = 0. The ab o v e construction pro vides the the analytic contin uation of (3.1). W e apply these constructions to G n ( z ; d 1 , d 1 ) = e i 2 d 1 √ z − ( πn/d 2 ) 2 i p z − ( π n/d 2 ) 2 − 1 i p z − ( π n/d 2 ) 2 , where w e use the same notation for the analytic con tinuation to the lo wer half-plane. Assume that z ∈ B ( ξ l,k ) and employ a sp ecial decomposition that will pla y an essential role in further discussion: K ε ( z ) = K 0 ( z ) + H ε ( z ) , (3.3) where K 0 ( z ) = ∞ X n =1 G n ( z ; d 1 , d 1 ) P 0 n , H ε ( z ) := ∞ X n =1 G n ( z )( P ε n − P 0 n ) , (3.4) and P 0 n = ( e 0 n , · ) L 2 ( I 0 ) e 0 n and P ε n = ( e ε n , · ) L 2 ( I 0 ) e ε n , i.e. all the op erators contributing (3.4) are defined in the space L 2 ( I 0 ); see discussion after form ula (3.1). Lemma 3.1 The op er ator H ε ( z ) is b ounde d and its norm admits the fol lowing asymptotics ∥ H ε ( z ) ∥ L 2 ( I 0 ) → L 2 ( I 0 ) = O ( ε 1 / 2 ) , (3.5) that is uniform for al l z ∈ B ( ξ k,l ) . 8 Pro of. All norms and scalar pro ducts in the follo wing proof will b e understo od in the space L 2 ( I 0 ). F or simplicit y , we omit explicit reference to the space whenever there is no confusion. Note that employing the decomp osition e 0 n = e ε n + ˘ e ε n , see (3.2), to P 0 n = ( e 0 n , · ) e 0 n yields P ε n f − P 0 n f = − ( ˘ e ε n , f ) e ε n − ( e ε n , f ) ˘ e ε n − ( ˘ e ε n , f ) ˘ e ε n . (3.6) Applying this to ( f , H ε ( z ) f ), we get | ( f , H ε ( z ) f ) | ≤ ∞ X n =1 | G n ( z ) | h | ( f , ˘ e ε n ) ( e ε n , f ) | + | ( f , e ε n ) ( ˘ e ε n , f ) | + | ( f , ˘ e ε n ) ( ˘ e ε n , f ) | i . (3.7) T o pro ceed further, we b egin by estimating the first of the three terms in the abov e b ound, namely T := ∞ X n =1 | G n ( z ) | | ( f , ˘ e ε n ) ( e ε n , f ) | . (3.8) Applying the Cauch y–Sch w artz inequality we get | ( f , ˘ e ε n ) | ≤ Z I 0 \ I ε | f ˘ e ε n | d x ≤ ε 1 / 2  Z I 0 \ I ε | f ˘ e ε n | 2 d x  1 / 2 ≤ ε 1 / 2 ∥ f ∥ , (3.9) where, in the last estimate, we used the fact that sup | ˘ e ε n | = 1. Note that for z ∈ B ( ξ l,k ) and ξ l,k  = ( π n/d 2 ) 2 w e can estimate | G n ( z ) | = | e i 2 d 1 √ z − ( πn/d 2 ) 2 − 1 | | p z − ( π n/d 2 ) 2 | ≤ C 1 n , (3.10) where the constant C dep ends on l , k, d 2 . 2 On the other hand, if there exists n such that ξ l,k = ( π n/d 2 ) 2 then we hav e asymptotics | G n ( z ) | = O ( δ ), where and δ = − ξ l,k + z . The latter can o ccur for one comp onen t n only , therefore (3.10) holds for any n ∈ N . Com bining (3.9) and (3.10) leads to T ≤ C ε 1 / 2 ∥ f ∥ ∞ X n =1 1 n | ( e ε n , f ) | ≤ C ′ ε 1 / 2 ∥ f ∥  ∞ X n =1 | ( f , e ε n ) | 2  1 / 2 , (3.11) where C ′ = C  P ∞ n =1 1 n 2  1 / 2 and w e again used the Cauch y–Sc hw artz inequalit y . Note that ∞ X n =1 | ( f , e ε n ) | 2 ≤ ∞ X n =1 | ( f , e 0 n ) | 2 = ∥ f ∥ 2 . Applying this to (3.11) w e obtain T ≤ C ε 1 / 2 ∥ f ∥ 2 . The remaining comp onen ts of (3.7) can b e estimated by the rep eating the argument. This yields the asymp- totics | ( f , ( H ε ( z ) f ) L 2 ( I 0 ) | = O ( ε 1 / 2 ) , (3.12) that is uniform with respect to f and, consequently , ∥ H ε ( z ) ∥ L 2 ( I 0 ) → L 2 ( I 0 ) = O ( ε 1 / 2 ). 2 In the following discussion, we will use the sym b ol C to denote a constan t, which may vary from line to line. The dependence of such constants on parameters will b e explicitly indicated, and any uniformity will b e stated. 9 Remark 3.2 F or fixed z (a wa y from the thresholds z = ( π n/d 2 ) 2 ) w e ha ve K 0 ( z ) e 0 n = λ n ( z ) e 0 n , λ n ( z ) := G n ( z ; d 1 , d 1 ) . Relying on (3.10) we conclude that K 0 ( z ) is compact, since it is diagonal in { e 0 n } ∞ n =1 with eigen v alues → 0. Therefore every nonzero eigenv alue (or an y finite set of eigenv alues) is isolated in σ ( K 0 ( z )), see [18, Ch. II I, § 6]. Fix n suc h that λ n ( z )  = 0 and assume first that λ n ( z ) is simple. Then δ n ( z ) := dist( λ n ( z ) , σ ( K 0 ( z )) \ { λ n ( z ) } ) > 0. Cho osing υ ∈ (0 , δ n ( z ) / 2) and Γ n := { λ : | λ − λ n ( z ) | = υ } gives a con tour Γ n ⊂ ρ ( K 0 ( z )) enclosing only λ n ( z ). Since K 0 ( z ) is normal, ∥ ( K 0 ( z ) − λ ) − 1 ∥ = 1 dist( λ, σ ( K 0 ( z ))) ≤ 1 υ , λ ∈ Γ n , (3.13) [18, Ch. II I, § 6]. In the case of a finite degeneracy , one encloses the whole finite cluster and the corresponding Riesz pro jector equals the sum of the inv olv ed rank-one pro jectors. Lemma 3.3 L et I 0 ⊂ R and c onsider the family of b ounde d op er ators K ε ( z ) = K 0 ( z ) + H ε ( z ) on L 2 ( I 0 ) , define d by (3.3) and (3.4) for z ∈ B ( ξ l,k ) and ε ≥ 0 . Then the fol lowing statements hold for al l z ∈ B ( ξ l,k ) and al l 0 < ε ≤ ε 1 for some ε 1 ∈ (0 , ε 0 ] : (1) Γ n lies in the r esolvent set of K ε ( z ) , and the R iesz pr oje ctors P K ε n ( z ) = − 1 2 π i Z Γ n ( K ε ( z ) − λ ) − 1 dλ, P 0 n ( z ) = P 0 n = − 1 2 π i Z Γ n ( K 0 ( z ) − λ ) − 1 dλ, ar e wel l-define d. (2) The pr oje ctors satisfy the estimate   P K ε n ( z ) − P 0 n   L 2 ( I 0 ) → L 2 ( I 0 ) ≤ | Γ n | 2 π ∥ H ε ( z ) ∥ L 2 ( I 0 ) → L 2 ( I 0 ) υ  υ − ∥ H ε ( z ) ∥ L 2 ( I 0 ) → L 2 ( I 0 )  . (3.14) (3) In p articular,   P K ε n ( z ) − P 0 n   L 2 ( I 0 ) → L 2 ( I 0 ) = O ( ε 1 / 2 ) as ε → 0 . (3.15) Pro of. Fix z and write K 0 := K 0 ( z ) and H ε := H ε ( z ). F or λ ∈ Γ n w e hav e K ε − λ = (I + H ε ( K 0 − λ ) − 1 )( K 0 − λ ) . By (3.13) w e ha ve sup λ ∈ Γ n ∥ ( K 0 − λ ) − 1 ∥ L 2 ( I 0 ) → L 2 ( I 0 ) ≤ 1 υ . and consequen tly ∥ H ε ( K 0 − λ ) − 1 ∥ L 2 ( I 0 ) → L 2 ( I 0 ) ≤ ∥ H ε ∥ L 2 ( I 0 ) → L 2 ( I 0 ) υ . Therefore for ε small enough the righ t-hand side is smaller than 1, cf. (3.5), and I + H ε ( K 0 − λ ) − 1 is in vertible. Moreo ver, ( K ε − λ ) − 1 = ( K 0 − λ ) − 1  I + H ε ( K 0 − λ ) − 1  − 1 . 10 Using the Neumann series,   (I + H ε ( K 0 − λ ) − 1 ) − 1   L 2 ( I 0 ) → L 2 ( I 0 ) ≤ υ υ − ∥ H ε ∥ L 2 ( I 0 ) → L 2 ( I 0 ) , and therefore ∥ ( K ε − λ ) − 1 ∥ L 2 ( I 0 ) → L 2 ( I 0 ) ≤ 1 υ − ∥ H ε ∥ L 2 ( I 0 ) → L 2 ( I 0 ) . (3.16) The resolv ent identit y gives ( K ε − λ ) − 1 − ( K 0 − λ ) − 1 = − ( K ε − λ ) − 1 H ε ( K 0 − λ ) − 1 , hence using (3.13) and (3.16),   ( K ε − λ ) − 1 − ( K 0 − λ ) − 1   L 2 ( I 0 ) → L 2 ( I 0 ) ≤ ∥ H ε ∥ L 2 ( I 0 ) → L 2 ( I 0 ) υ ( υ − ∥ H ε ∥ L 2 ( I 0 ) → L 2 ( I 0 ) ) . In tegrating ov er Γ n pro ves (3.14), and using (3.5) yields (3.15). Corollary 3.4 Assume that the sp e ctr al cluster of K 0 ( z ) inside Γ n c onsists of a simple eigenvalue, so that rank P 0 n = 1 . L et e 0 n b e the c orr esp onding normalize d eigenfunction. F or ε > 0 sufficiently smal l, rank P K ε n ( z ) = 1 as wel l. L et e K ε n ( z ) denote the normalize d eigenfunction sp anning Ran P K ε n ( z ) , chosen so that  e K ε n ( z ) , e 0 n  L 2 ( I 0 ) ∈ R + . (3.17) Then ther e exist o ε = o (1) and a ve ctor η ε n ( z ) ⊥ e 0 n such that e K ε n ( z ) = (1 + o ε ) e 0 n + η ε n ( z ) , (3.18) and ∥ η ε n ( z ) ∥ L 2 ( I 0 ) = O ( ε 1 / 2 ) , as ε → 0 . (3.19) Pro of. Define α ε :=  e K ε n ( z ) , e 0 n  L 2 ( I 0 ) ∈ R + , η ε n ( z ) := e K ε n ( z ) − α ε e 0 n . By construction, η ε n ( z ) ⊥ e 0 n . Since ∥ e K ε n ( z ) ∥ L 2 ( I 0 ) = ∥ e 0 n ∥ L 2 ( I 0 ) = 1, w e ha ve the orthogonal decomposition 1 = ∥ e K ε n ( z ) ∥ 2 L 2 ( I 0 ) = α 2 ε + ∥ η ε n ( z ) ∥ 2 L 2 ( I 0 ) . (3.20) Recall that, with our con ven tion on the inner pro duct, P 0 n u = ( e 0 n , u ) L 2 ( I 0 ) e 0 n , P K ε n ( z ) u = ( e K ε n ( z ) , u ) L 2 ( I 0 ) e K ε n ( z ) . Therefore, ( P K ε n ( z ) − P 0 n ) e 0 n = ( e K ε n ( z ) , e 0 n ) L 2 ( I 0 ) e K ε n ( z ) − e 0 n = α ε e K ε n ( z ) − e 0 n . T aking the L 2 ( I 0 )–norm and using Lemma 3.3, we obtain ∥ α ε e K ε n ( z ) − e 0 n ∥ L 2 ( I 0 ) = ∥ ( P K ε n ( z ) − P 0 n e 0 n ∥ L 2 ( I 0 ) ≤ ∥ P K ε n ( z ) − P n ( z ) ∥ L 2 ( I 0 ) → L 2 ( I 0 ) = O ( ε 1 / 2 ) . (3.21) On the other hand, using (3.20), ∥ α ε e K ε n ( z ) − e 0 n ∥ 2 L 2 ( I 0 ) = α 2 ε + 1 − 2 α 2 ε = 1 − α 2 ε = ∥ η ε n ( z ) ∥ 2 L 2 ( I 0 ) . Hence ∥ η ε n ( z ) ∥ L 2 ( I 0 ) = ∥ ( P K ε n ( z ) − P 0 n ) e 0 n ∥ L 2 ( I 0 ) = O ( ε 1 / 2 ) , (3.22) 11 whic h prov es (3.19). Finally , since α ε ∈ [0 , 1] and o ε = 1 − α ε , w e ha ve | o ε | = | 1 − α ε | ≤ 1 − α 2 ε = ∥ η ε n ( z ) ∥ 2 L 2 ( I 0 ) = O ( ε ) , and therefore o ε = o (1) as ε → 0. This completes the proof. With these results, the problem of k er K ε ( z ) can b e transferred to the problem of solving the following scalar equation ζ n ( z , ε ) = 0 , where ζ n ( z , ε ) :=  e K ε n ( z ) , K ε ( z ) e K ε n ( z )  L 2 ( I 0 ) . (3.23) Remark 3.5 Although the parameter ε is ph ysically restricted to ε > 0, in the analysis b elo w w e consider a con tinuous extension of the function ζ k ( z , ε ) to ε in a neighborho od of 0 in R . This extension is used solely as a technical device in order to apply the implicit function theorem; the resulting solution branch is subsequen tly restricted to ε > 0. The follo wing theorem establishes the existence of a solution to equation (3.23). Lemma 3.6 Ther e exists ε 0 > 0 and a uniquely define d, c ontinuous function ε 7→ z ( ε ) , ε ∈ [0 , ε 0 ) such that the e quation ζ k ( z ( ε ) , ε ) = 0 (3.24) holds. Pro of. The first observ ation sho ws that ζ k ( ξ l,k , 0) =  e 0 k , K 0 ( ξ l,k ) e 0 k  L 2 ( I 0 ) = 0 . (3.25) F urthermore, employing the deriv ativ e formula d dz G n ( z ; d 1 , d 1 ) = d 1 e 2 id 1 √ z − (( πn/d 2 ) 2 z − ( π n/d 2 ) 2 + i ( e 2 ia √ z − ( πn/d 2 ) 2 − 1) 2( z − ( π n/d 2 ) 2 ) 3 / 2 , (3.26) and the form of K 0 ( z ), we state that d dz ζ k ( z , ε )   z = ξ l,k , ε =0 = " d 1 e 2 id 1 √ ξ l,k − ( π n/d 2 ) 2 ξ l,k − ( π n/d 2 ) 2 + i ( e 2 id 1 √ ξ l,k − ( π n/d 2 ) 2 − 1) 2( ξ l,k − ( π n/d 2 ) 2 ) 3 / 2 #  e 0 k , e 0 k  L 2 ( I 0 ) = d 3 1 π 2 , (3.27) i.e. d dz ζ k ( z , ε )   z = ξ l,k , ε =0  = 0 . (3.28) F urthermore, rep eating the arguments from the proof of Lemma 3.1, one can sho w that ˝ BA CK   ( f , ( K ε ( z ) − K ε ′ ( z )) f ) L 2 ( I 0 )   ≤ C | ε − ε ′ | 1 / 2 ∥ f ∥ 2 L 2 ( I 0 ) (3.29) for an y f ∈ L 2 ( I 0 ). This implies that ε 7→ ζ k ( z , ε ) is a con tinuous function. Relying on the fact that z 7→ K ε ( z ) is an analytic op erator-v alued function for z ∈ B ( ξ l,k ), we can conclude that z 7→ ζ k ( z , ε ) is analytic in B ( ξ l,k ) for any ε . W e extend the function ζ k ( z , ε ) to ε ∈ ( − ε 0 , ε 0 ), see Remark 3.5. Combining the ε -con tinuit y and z -analyticit y with (3.25) and (3.28), and applying the Contin uous Implicit F unction Theorem, (see, for example [17, Thm. 2]) we state that there exists a unique contin uous function ε 7→ z ( ε ) suc h that ζ k ( z ( ε ) , ε ) = 0 , for ε small enough. This completes the proof. 12 4 Asymptotic analysis of resonances In this section, we inv estigate the asymptotic b eha vior of the function ε 7→ z ( ε ), whic h satisfies equa- tion (3.24). T o deriv e this asymptotics, we introduce an auxiliary function δ ( ε ), defined such that z ( ε ) = ξ l,k + δ ( ε ) , (4.1) and analyze The corrections follo ws from the fact that o ε is selected real ζ k ( z , ε ) =  e K ε k ( ξ l,k + δ ( ε )) , K ε ( z ( ε )) e K ε k ( ξ l,k + δ ( ε ))  L 2 ( I 0 ) =(1 + o 2 ε )  e 0 k , K 0 ( z ( ε )) e 0 k  L 2 ( I 0 ) + (1 + o ε )  e 0 k , K 0 ( z ( ε )) η ε k ( z ( ε ))  L 2 ( I 0 ) + (1 + o ε )  η ε k ( z ( ε )) , K 0 ( z ( ε )) e 0 k  L 2 ( I 0 ) +  η ε k ( z ( ε )) , K 0 ( z ( ε )) η ε k ( z ( ε ))  L 2 ( I 0 ) +  e K ε k ( z ( ε )) , H ε ( z ( ε )) e K ε k ( z ( ε ))  L 2 ( I 0 ) . In the abov e, w e used the fact that o ε is chosen to be real. Using the fact that K 0 ( z ( ε )) e 0 k remains in the space spanned by e 0 k and η ε k ( z ( ε )) is orthogonal to e 0 k w e come to the conclusion ζ k ( z , ε ) = (1 + o 2 ε )  e 0 k , K 0 ( z ( ε )) e 0 k  L 2 ( I 0 ) + J ε , (4.2) where P J ε :=  η ε k ( z ( ε )) , K 0 ( z ( ε )) η ε k ( z ( ε ))  L 2 ( I 0 ) +  e K ε k ( z ( ε )) , H ε ( z ( ε )) e K ε k ( z ( ε ))  L 2 ( I 0 ) , (4.3) whic h, in view of (3.17), can be developed as J ε = (1 + o 2 ε )  e 0 k , H ε ( z ( ε )) e 0 k  L 2 ( I 0 ) + (1 + o ε )  e 0 k , H ε ( z ( ε )) η ε k ( z ( ε ))  L 2 ( I 0 ) , (4.4) where w e used again orthogonality of η ε k and e 0 k , the fact that ξ k ( z ( ε ) , ε ) is an eigenv alue of K ε ( z ) to e K ε k ( z ) whic h ξ k ( z , ε ) = 0 for (4.1). Employing the asymptotics (3.5) and (3.22), we state J ε = O ( ε 1 / 2 ) . Ho wev er, in the follo wing discussion, we refine the asymptotic behavior of J ε b y analyzing eac h term in equation (4.4). The first term is estimated in the lemma b elo w. Lemma 4.1 F or z ∈ B ( ξ l,k ) the fol lowing asymptotics    e 0 k , ( H ε ( z ) e 0 k  L 2 ( I 0 )   = O ( ε 2 ) , (4.5) holds. Pro of. Using the decomp osition (3.4) we hav e  e 0 m , H ε ( z ) e 0 m  L 2 ( I 0 ) = ∞ X n =1 G n ( z ; d 1 , d 1 )  | ( e ε n , e 0 m ) L 2 ( I 0 ) | 2 − δ nm  (4.6) = ∞ X n =1 G n ( z ; d 1 , d 1 ) | ( e ε n , e 0 m ) L 2 ( I 0 ) | 2 − G m ( z ; d 1 , d 1 ) . (4.7) Therefore    e 0 m , H ε ( z ) e 0 m  L 2 ( I 0 )   ≤ ∞ X n =1 n  = m | e i 2 d 1 √ z − ( πn/d 2 ) 2 − 1 | | p z − ( π n/d 2 ) 2 | | ( e ε n , e 0 m ) L 2 ( I 0 ) | 2 + | e i 2 d 1 √ z − ( πm/d 2 ) 2 − 1 | | p z − ( π m/d 2 ) 2 |   | ( e ε m , e 0 m ) L 2 ( I 0 ) | 2 − 1   . (4.8) 13 A direct calculation shows that for n  = m w e hav e    e ε n , e 0 m  L 2 ( I 0 )   2 = 2 d 2  Z I 0 sin  π nx 2 d 2  χ I ε ( x 2 ) sin  π mx 2 d 2  d x 2  2 = 2 d 2  Z I ε sin  π nx 2 d 2  sin  π mx 2 d 2  d x 2  2 = 2 d 2 π 2   cos  ( n − m ) π (2 t + ε ) 2 d 2  sin  ( n − m ) π ε 2 d 2  n − m − cos  ( n + m ) π (2 t + ε ) 2 d 2  sin  ( n + m ) π ε 2 d 2  n + m   2 ≤ 4 d 2 π 2   cos 2  ( n − m ) π (2 t + ε ) 2 d 2  sin 2  ( n − m ) π ε 2 d 2  ( n − m ) 2 + cos 2  ( n + m ) π (2 t + ε ) 2 d 2  sin 2  ( n + m ) π ε 2 d 2  ( n + m ) 2   . The expressions corresponding to the ab o ve sum will b e denoted as I n − m and I n + m F urthermore, for z ∈ B ( ξ l,k ) w e estimate ∞ X n =1 n  = m | e i 2 d 1 √ z − ( πn/d 2 ) 2 − 1 | | p z − ( π n/d 2 ) 2 | I n − m = ∞ X n =1 n  = m | e i 2 d 1 √ z − ( πn/d 2 ) 2 − 1 | | p z − ( π n/d 2 ) 2 | 1 ( n − m ) 2 cos 2  ( n − m ) π (2 t + ε ) 2 d 2  sin 2  ( n − m ) π ε 2 d 2  ≤ ∞ X n =1 n  = m | e i 2 d 1 √ z − ( πn/d 2 ) 2 − 1 | | p z − ( π n/d 2 ) 2 | 1 ( n − m ) 2 sin 2  ( n − m ) π ε 2 d 2  = ∞ X w = − m +1 w  =0 | e i 2 d 1 √ z − ( π ( w + m ) /d 2 ) 2 − 1 | | p z − ( π ( w + m ) /d 2 ) 2 | 1 w 2 sin 2  w πε 2 d 2  . Pro ceeding in the same w a y as in the pro of of Lemma 3.1, w e show that there exists a constan t C , dep ending on w, m and d 2 , suc h that | e i 2 d 1 √ z − ( π ( w + m ) /d 2 ) 2 − 1 | | p z − ( π ( w + m ) /d 2 ) 2 | ≤ C 1 | w + m | , for w ≥ 1 − m , for z ∈ B ( ξ l,k ); see (3.10). This enables us to establish the b ound ∞ X w = − m +1 w  =0 | e i 2 d 1 √ z − ( π ( w + m ) /d 2 ) 2 − 1 | | p z − ( π ( w + m ) /d 2 ) 2 | 1 w 2 sin 2  w πε 2 d 2  ≤ C ∞ X w = − m +1 w  =0 1 ( w + m ) 1 w 2 sin 2  w πε 2 d 2  . The last sum can decomposed on to P − 1 w = − m +1 and P ∞ w =1 . In the first case w e ha ve the following asymptotics − 1 X w = − m +1 1 ( w + m ) w 2 sin 2  w πε 2 d 2  = O ( ε 2 ) , (4.9) whic h is not uniform with resp ect to m 3 , and in the second case, we get ∞ X w =1 1 ( w + m ) w 2 sin 2  w πε 2 d 2  ≤ ∞ X w =1 1 w 3 sin 2  w πε 2 d 2  = ε 3 h X w ′ ∈{ ε, 2 ε,... } 1 w ′ 3 sin 2  w ′ π 2 d 2  i = ε 3 h X w ′ ∈{ ε, 2 ε, 3 ε...N 0 } 1 w ′ 3 sin 2  w ′ π 2 d 2  + X w ′ ∈{ N 0 + ε,N 0 +2 ε... } 1 w ′ 3 sin 2  w ′ π 2 d 2  i , (4.10) 3 Additional remarks are given in the following discussion. 14 where N 0 is a p ositiv e n umber. The first sum in the brac ket can be b ounded by the num b er of comp onen ts N 0 ε . The second sum in the brack et can b e b ounded by a constant undep enden t of ε . Therefore ∞ X n =1 n  = m | e i 2 d 1 √ z − ( πn/d 2 ) 2 − 1 | | p z − ( π n/d 2 ) 2 | I n − m = O ( ε 2 ) . In the same wa y we show ∞ X n =1 n  = m | e i 2 d 1 √ z − ( πn/d 2 ) 2 − 1 | | p z − ( π n/d 2 ) 2 | I n + m = O ( ε 2 ) . F urthermore, an explicit calculus sho ws that | ( e ε m , e 0 m ) L 2 ( I 0 ) | 2 − 1 = O ( ε 2 ) . Consequen tly all terms in (4.8) b eha v e as O ( ε 2 ). The ab o v e result sho ws that the first term in (4.4) given by (1 + o 2 ε )  e 0 k , H ε ( z ( ε )) e 0 k  L 2 ( I 0 ) b eha v es as O ( ε 2 ), since o ε = o (1). The next statemen t provides the estimates for the remaining term of J ε from (4.4). Lemma 4.2 We have    e 0 k , H ε ( z ( ε )) η ε k ( z ( ε ))  L 2 ( I 0 )   = O ( ϵ 2 ) . (4.11) Pro of. In order to pro ve the ab o ve asymptotics, w e write η ε k ( z ) by means of the basis { e 0 n } ∞ k =1 , i.e. η ε k ( z ( ε )) = ∞ X n =1 n  = k a n ( z ( ε )) e 0 n . This, in view of (3.10), yields    e 0 k , H ε ( z ( ε )) η ε k ( z ( ε ))  L 2 ( I 0 )   ≤ C ∞ X n =1 n  = k | a n ( z ( ε )) | ∞ X n ′ =1 1 n ′ h ( e ε n ′ , e 0 k ) L 2 ( I 0 ) ( e 0 n , e ε n ′ ) L 2 ( I 0 ) − δ kn ′ δ n ′ n i = C ( L 1 + L 2 ) (4.12) where L 1 = ∞ X n =1 n  = k | a n ( z ( ε )) | ∞ X n ′ =1 n ′  = n 1 n ′   ( e ε n ′ , e 0 k ) L 2 ( I 0 ) ( e 0 n , e ε n ′ ) L 2 ( I 0 )   , and L 2 = ∞ X n =1 n  = k | a n ( z ( ε )) | 1 n   ( e ε n , e 0 k ) L 2 ( I 0 ) ( e 0 n , e ε n ) L 2 ( I 0 )   . The first term can be estimated as L 1 ≤  ∞ X n =1 n  = k | a n ( z ( ε )) | 2  1 / 2 · ∞ X n ′ =1 n ′  = n h 1 n ′    e 0 k , e ε n ′ ) L 2 ( I 0 )   ·  ∞ X n =1 n  = k   ( e ε n ′ , e 0 n ) L 2 ( I 0 )   2  1 / 2 i (4.13) Note that  ∞ X n =1 n  = k | a n ( z ( ε )) | 2  1 / 2 = ∥ η ε k ( z ( ε )) ∥ L 2 ( I 0 ) = O ( ε 1 / 2 ) , (4.14) 15 cf. (3.22). F urthermore, repeating the argumen ts from the proof of Lemma 4.1 we state that ˜ L 1 := ∞ X n =1 n  = n ′   ( e ε n ′ , e 0 n ) L 2 ( I 0 )   2 ≤ C ∞ X w = − n ′ +1 1 w 2 sin 2  w πε 2 d 2  = O ( ε ) , (4.15) where we use the reasoning analogous to (4.9) and (4.10), with the difference that here the relev ant denomi- nator contains the second p o wer instead of the third one; consequen tly , the resulting asymptotics is of order O ( ε ) rather than O ( ε 2 ). This implies L 1 ≤ ˜ L 1 / 2 1 ∥ η ε k ( z ( ε )) ∥ L 2 ( I 0 ) ·  ∞ X n ′ =1 n ′  = k 1 n ′    e 0 k , e ε n ′ ) L 2 ( I 0 )    . (4.16) F urthermore, applying the analysis analogous to that from the pro of of Lemma 4.1 we state ∞ X n ′ =1 n ′  = k 1 n ′   ( e 0 k , e ε n ′ ) L 2 ( I 0 )   = O ( ε ) . (4.17) Com bining (4.14), (4.15), and (4.17) and substituting them into (4.13), we obtain L 1 = O ( ε 2 ). The term L 2 can be estimated in the same w ay and satisfies the same b ound. Therefore, (4.11) follo ws. F rom the ab o ve lemma w e conclude that (1 + o ε )  e 0 k , H ε ( z ( ε )) η ε k ( z ( ε ))  L 2 ( I 0 ) app earing in (4.4) b eha ves as O ( ε 2 ). Summarizing the statements of Lemmae 4.1 and 4.2, and inserting them into (4.3), we obtain, in view of (4.2), ζ k ( z , ε ) = (1 + o 2 ε )  e 0 k , K 0 ( ξ k,l + δ ( ε )) e 0 k  L 2 ( I 0 ) + J ε , where J ε = O ( ε 2 ) . (4.18) A t this stage we are ready to form ulate and finish the main result of this section. Theorem 4.3 Assume that ξ l,k is a simple eigenvalue of − ∆ D ( C ) . Then ther e exists a uniquely determine d function ε 7→ z ( ε ) , ε ∈ (0 , ε 0 ) such that k er K ε  z ( ε )   = ∅ , with the fol lowing asymptotic exp ansion z ( ε ) = ξ l,k + µ ( ε ) + i ν ( ε ) , (4.19) wher e the r e al-value d functions µ ( ε ) and ν ( ε ) satisfy µ ( ε ) = O ( ε 2 ) and ν ( ε ) = O ( ε 2 ) . Now, supp ose that ξ l,k has multiplicity N . In this c ase, ther e exist N functions ε 7→ z j ( ε ) , for j = 1 , . . . , N , such that k er K ε  z j ( ε )   = ∅ , with e ach function admitting the asymptotic exp ansion z j ( ε ) = ξ l,k + µ j ( ε ) + i ν j ( ε ) , wher e the c orr esp onding functions µ j ( ε ) and ν j ( ε ) b ehave as O ( ε 2 ) . Pro of. Assume first that ξ l,k is a simple eigenv alue. Then there exists a unique contin uous function ε 7→ z ( ε ) suc h that ζ k ( z ( ε ) , ε ) = 0 and z (0) = ξ l,k , cf. Lemma 3.6. Assuming now that z = ξ l,k + δ we analyze the first component of (4.18), namely  e 0 k , K 0 ( ξ l,k + δ ) e 0 k  L 2 ( I 0 ) = G k ( ξ l,k + δ ; d 1 , d 1 ) = e 2 d 1 i √ ξ l,k + δ − ( πk/d 2 ) 2 − 1 i p ξ l,k + δ − ( πk /d 2 ) 2 = e 2 d 1 i √ δ +( πl/d 1 ) 2 − 1 i p δ + ( π l/d 1 ) 2 , (4.20) 16 where w e again used the fact that { e 0 n } ∞ n =1 form the orthonormal basis. Applying no w the T aylor expansion of δ 7→ √ δ + ω 2 = ω + 1 2 ω δ + O ( δ 2 ) to the function g ( δ ) := i √ δ + ω 2  e 2 d 1 i √ δ + ω 2 − 1  , w e get g ( δ ) = i ω  e 2 d 1 iω − 1  + δ  − d 1 e 2 d 1 iω ω 2 − i ( e 2 d 1 iω − 1) 2 ω 3  + O ( δ 2 ) . Setting ω = π l d 1 , w e conclude from (4.20) that  e 0 k , K 0 ( ξ l,k + δ ) e 0 k  L 2 ( I 0 ) = − d 3 1 ( π l ) 2 δ + O ( δ 2 ) . Therefore the dominating comp onen t in the ab o ve expression b eha v es as − d 3 1 ( π l ) 2 δ . In view of (4.18) and taking in to accoun t the fact that ζ k ( z ( ε ) , ε ) = 0 and J ε = O ( ε 2 ), w e conclude that δ = O ( ε 2 ) . This completes the pro of of the first part of the theorem under the assumption that ξ l,k is simple. W e now consider the case where ξ l,k is degenerate with multiplicit y N. This means that there exists N pairs of integers ( l j , k j ) , ... , j = 1 , ..., N suc h that ξ l,k =  l 1 π d 1  2 +  k 1 π d 2  2 = ... =  l N π d 1  2 +  k N π d 2  2 (4.21) Let e 0 l j e 0 k j , j = 1 , ... N stand for the corresponding eigenfunctions. Then for any e 0 k j w e hav e ζ k j ( ξ l,k , 0) =  e 0 k j , K 0 ( ξ l,k ) e 0 k j  L 2 ( I 0 ) = G k j ( ξ l,k ; d 1 , d 1 ) = e 2 d 1 i √ ξ l,k − ( π k j /d 2 ) 2 − 1 i p ξ l,k − ( π k j /d 2 ) 2 . (4.22) Applying (4.21), we obtain ξ l,k −  π k j d 2  2 =  π l j d 1  2 , i.e., p ξ l,k − ( π k j /d 2 ) 2 = π l j /d 1 , whic h implies that ζ k j ( ξ l,k , 0) = 0 , j = 1 , . . . , N . (4.23) This means that there exist N functions ζ k j ( z , ε ) =  e K ε k j ( z ) , K ε ( z ) e K ε k j ( z )  L 2 ( I 0 ) fulfilling (4.23). Employing again Lemma 3.6 we conclude that there exist N contin uous functions z j ( ε ) satisfying ζ k j ( z j ( ε ) , ε ) = 0 . The asymptotic b eha vior of z j ( ε ) can b e studied b y repeating the arguments used in the discussion of the simple pole case. Remark 4.4 There arises the question: how the components µ ( ε ) and ν ( ε ) in (4.19) depend on the quan tum n umbers l and k . It should b e emphasized that the reasoning developed in Lemma 4.1 shows that the asymptotics in equation (4.19) are not uniform with respect to k . This non uniformit y follows, for example, from the asymptotics in equation (4.9), which b eha ve lik e log m for large m . Consequently , it affects (4.5), (4.4) and (4.19). 17 Remark 4.5 While the leading terms of b oth the real and imaginary parts of the resonance p ole are generi- cally quadratic in the relativ e windo w measure, obtaining an explicit form ula for the corresponding coefficient in terms of the em b edded–eigen v alue eigenfunctions is substan tially more delicate. In small–aperture cou- pling problems the leading co efficien t is typically gov erned b y the trace of the unp erturbed mo de on the op ening. If this trace v anishes, for example b ecause the eigenfunction has a no de on the window or due to a symmetry constraint then the leading term cancels and the resonance width is determined b y higher order effects. Therefore, if the eigenfunction corresp onding to the em b edded eigenv alue v anishes on the windo w, the O ( ε 2 ) estimate is in general not exp ected to b e optimal. A detailed analysis of this situation is b ey ond the scope of the present pap er. 5 Resonances in higher dimensional systems: A study of three- dimensional w a veguides 5.1 Resonances induced by a rectangular gap in a w a veguide In this section we study a three-dimensional wa v eguide characterized by width d 2 and heigh t d 3 : Σ := { ( x 1 , x 2 , x 3 ) : x 1 ∈ [0 , ∞ ) , x 2 ∈ [ 0 , d 2 ] , x 3 ∈ [ 0 , d 3 ] } . Define ¯ I ε := { ( d 1 , x 2 , x 3 ) , x 2 ∈ ¯ I 2 ,ε 2 = [ t 2 , t 2 + ε 2 ] , x 3 ∈ ¯ I 3 ,ε 3 = [ t 3 , t 3 + ε 3 ] } , where ε i ≥ 0, t i ≥ 0, t i + ε i ≤ d i for i = 2 , 3. In the following we assume that ε = ε 2 = ε 3 a , (5.1) where a is a positive constant. Then I ε := I 0 \ ¯ I ε , where I 0 = { ( d 1 , x 2 , x 3 ) : x 2 ∈ [0 , d 2 ] , x 3 ∈ [0 , d 3 ] } . In the follo wing we will also use I j,ε j = I j, 0 \ ¯ I j,ε j . W e define the Hamiltonian in the analogous w ay as b efore − ∆ D I ε : W 2 , 2 0 (Σ \ I ε ) → L 2 (Σ) . Its resolv ent R ( z ) := ( − ∆ D − z ) − 1 is no w defined as an integral op erator with the kernel G ( z ; ( x 1 , x 2 , x 3 ) , ( y 1 , y 2 , y 3 )) := ∞ X k 2 =1 ∞ X k 3 =1 G k 2 ,k 3 ( z ; x 1 , y 1 ) e 0 k 2 ( x 2 ) e 0 k 2 ( y 2 ) e 0 k 3 ( x 3 ) e 0 k 3 ( y 3 ) , (5.2) where G k 2 ,k 3 ( z ; x 1 , y 1 ) = e iτ k 2 ,k 3 ( z ) | x 1 + y 1 | iτ k 2 ,k 3 ( z ) − e iτ k 2 ,k 3 ( z ) | x 1 − y 1 | iτ k 2 ,k 3 ( z ) , (5.3) with τ k 2 ,k 3 ( z ) := p z − ( π k 2 /d 2 ) 2 − ( π k 3 /d 3 ) 2 and ℑ τ k 2 ,k 3 ( z ) > 0, k 2 , k 3 ∈ N . The corresp onding embedded eigen v alues tak e the form ξ k 1 ,k 2 ,k 3 =  π k 1 d 1  2 +  π k 2 d 2  2 +  π k 3 d 3  2 . (5.4) W e define e 0 n j = s 2 d j sin  π n j x j d j  , j = 1 , 2 , 3 . and for j = 2 , 3 we introduce notation e ε j n j = e 0 n j χ I ε j , and the corresp onding pro jectors P ε j n j = ( e ε j n j , · ) L 2 ( I ε j ) e ε j n j . 18 F urthermore, we define the k ey op erator K ε ( z ) = ∞ X n 2 =1 ∞ X n 3 =1 G n 2 ,n 3 ( z ; d 1 , d 1 ) P ε 2 n 2 P ε 3 n 3 : L 2 ( I 0 ) → L 2 ( I 0 ) , (5.5) In analogy to the t wo-dimensional mo del, we employ the decomp osition K ε ( z ) = K 0 ( z ) + H ε ( z ) . The main result of this section establishes asymptotics O ( ε 4 ) of the p erturbativ e terms. Theorem 5.1 Assume that (5.1) holds and ξ k 1 ,k 2 ,k 3 has multiplicity N . Then ther e exist N c ontinuous functions ε 7→ z j ( ε ) , for j = 1 , . . . , N , such that k er K ε  z j ( ε )   = ∅ , with e ach function admitting the asymptotic exp ansion z j ( ε ) = ξ k 1 ,k 2 ,k 3 + µ j ( ε ) + i ν j ( ε ) , µ j ( ε ) = O ( ε 4 ) , ν j ( ε ) = O ( ε 4 ) . The strategy of pro of is based on the to ols dev elop ed in the previous section. Here, we fo cus on the essen tial differences. First we show the analog of Lemma 4.1. Lemma 5.2 Assume that z ∈ B ( ξ k 1 ,k 2 ,k 3 ) and assumption (5.1) is satisfie d. Then the asymptotics ∥ H ε ( z ) ∥ L 2 ( I 0 ) → L 2 ( I 0 ) = O ( ε 1 / 2 ) (5.6) holds. Remark 5.3 Let us note that the ab o ve asymptotics is weak er than the one established in Lemma 3.6, due to the fact that now the v olume of the gap b eha ves as | ¯ I ε | = O ( ε 2 ). Ho w ever, the asymptotics given in (5.6) is sufficien t at this stage and will b e strengthened in the subsequen t steps of the discussion. Pro of. In the course of this pro of, all scalar pro ducts and norms without an explicit space indicator are assumed to b e taken in L 2 ( I 0 ). T o show the statemen t w e observe | ( f , H ε ( z ) f ) | ≤ ∞ X n 2 =1 ∞ X n 3 =1 | G n 2 ,n 3 ( z ) | h | ( f , ˘ h ε n 2 ,n 3 f ) ( h ε n 2 ,n 3 , f ) | + | ( f , h ε n 2 ,n 3 f ) ( ˘ h ε n 2 ,n 3 , f ) | + | ( f , ˘ h ε n 2 ,n 3 ) ( ˘ h ε n 2 ,n 3 , f ) | i . (5.7) where ˘ h ε n 2 ,n 3 := ˘ e ε n 2 ˘ e aε n 3 and h ε n 2 ,n 3 := e ε n 2 e aε n 3 , we used (5.1). W e assume that f admits separation of v ariables, i.e., f ( x 2 , x 3 ) = f 2 ( x 2 ) f 3 ( x 3 ). Denote the first comp onen t of (5.7) as T = ∞ X n 2 =1 ∞ X n 3 =1 | G n 2 ,n 3 ( z ) | | ( f , ˘ h ε n 2 ,n 3 ) ( h ε n 2 ,n 3 , f ) | , whic h determines an analogue of (3.8). Relying on the se paration of v ariables of the function f and proceeding analogously to the pro of of Lemma 4.1, we obtain | ( f , ˘ h ε n 2 ,n 3 ) | ≤ ε 1 / 2  Z ¯ I 2 ,ε | f 2 ˘ e ε n 2 | 2 d x 2  1 / 2 · | ( f 3 , ˘ e aε n 3 ) L 2 ( I 3 , 0 ) | ≤ ε 1 / 2 ∥ f 2 ∥ L 2 ( I 2 , 0 ) · | f n 3 | , (5.8) where f n 3 = ( e 0 n 3 , f ) L 2 ( I 3 , 0 ) , and we used the estimate | ( e aε n 3 , f ) L 2 ( I 3 , 0 ) | ≤ | f n 3 | together with the Cauch y– Sc hw artz inequalit y . Similarly , we hav e | ( h ε n 2 ,n 3 , f ) | ≤ | f n 2 | · | f n 3 | . 19 F urthermore, mimicking the argument from the pro of of Lemma 4.1 yields | G n 2 ,n 3 ( z ) | ≤ C 1 p n 2 2 + n 2 3 . (5.9) Emplo ying again Cauch y–Sch wartz inequality and the fact that P ∞ n i =1 | f n i | 2 = ∥ f i ∥ 2 L 2 ( I i, 0 ) for i = 2 , 3, we summarize T ≤ C ε 1 / 2 ∥ f 2 ∥ L 2 ( I 2 , 0 ) ∞ X n 2 =1 ∞ X n 3 =1 1 p n 2 2 + n 2 3 | f n 2 | · | f n 3 | 2 ≤ C ε 1 / 2 ∥ f 2 ∥ L 2 ( I 2 , 0 )  ∞ X n 3 =1 | f n 3 | 2  ·  ∞ X n 2 =1 1 n 2 | f n 2 |  ≤ C ε 1 / 2 ∥ f 2 ∥ L 2 ( I 2 , 0 ) ∥ f 3 ∥ 2 L 2 ( I 3 , 0 )  ∞ X n 2 =1 | f n 2 | 2  1 / 2 ·  ∞ X n 2 =1 1 n 2 2  1 / 2 ≤ C ′ ε 1 / 2 ∥ f 2 ∥ 2 L 2 ( I 2 , 0 ) ∥ f 3 ∥ 2 L 2 ( I 3 , 0 ) = C ′ ε 1 / 2 ∥ f ∥ 2 . (5.10) The ab o ve inequalities can b e extended by the con tinuoit y to the whole space L 2 ( I 0 ). Estimating the remaining terms of (5.7) in the same w ay we state (5.1). Using the ab o v e result, we can rep eat the steps from the tw o-dimensional case and, with the help of the Contin uous Implicit F unction Theorem, establish the analogue of Lemma 3.6. That is, there exists a uniquely defined, contin uous function ε 7→ z ( ε ), defined for ε ∈ [0 , ε 0 ), ε 0 > 0, and suc h that the equation ζ k 2 ,k 3 ( z ( ε ) , ε ) = 0 (5.11) is satisfied, where ζ k 2 ,k 3 ( z , ε ) denotes the eigen v alue of K ε ( z ) corresp onding to the eigenfunction h K ε k 2 ,k 3 ( z ), whic h represents a perturbation of e 0 k 2 e 0 k 3 , i.e., h K ε k 2 ,k 3 ( z ) = (1 + o ε ) e 0 k 2 e 0 k 3 + η ε k 2 ,k 3 ( z ( ε )) , where ∥ η ε k 2 ,k 3 ( z ( ε )) ∥ L 2 ( I 0 ) = O ( ε 1 / 2 ). T o proceed further, we derive estimates analogous to (4.4), which in the presen t setting take the form J ε = (1 + o 2 ε )  e 0 k 2 e 0 k 3 , H ε ( z ( ε )) e 0 k 2 e 0 k 3  L 2 ( I 0 ) + (1 + o ε )  e 0 k 2 e 0 k 3 , H ε ( z ( ε )) η ε k 2 ,k 3 ( z ( ε ))  L 2 ( I 0 ) . (5.12) The analogue of Lemma 4.1 now takes the following form. Lemma 5.4 F or z ∈ B ( ξ k 1 ,k 2 ,k 3 ) the fol lowing asymptotics    e 0 k 2 e 0 k 3 , H ε ( z ) e 0 k 2 e 0 k 3 )  L 2 ( I 0 )   = O ( ε 4 ) (5.13) holds. Pro of. Similarly to before, we omit the explicit indication of the underlying space in scalar pro ducts and norms, assuming that the space is defined as L 2 ( I 0 ). The pro of is inspired by the reasoning developed in the pro of of Lemma 4.1. Here, we fo cus only on the subtle steps related to the fact that the current statemen t concerns the three-dimensional case. W e consider  e 0 m 2 e 0 m 3 , H ε ( z ) e 0 m 2 e 0 m 3  ≤ ∞ X n 2 =1 ∞ X n 3 =1 | G n 2 ,n 3 ( z ) |  | ( e ε n 2 , e 0 m 2 ) | 2 | ( e aε n 3 , e 0 m 3 ) | 2 − δ n 2 m 2 δ n 3 m 3  Because of that, we get:    e 0 m 2 e 0 m 3 , H ε ( z ) e 0 m 2 e 0 m 3    ≤ C  ∞ X n 2 =1 ∞ X n 3 =1 1 p n 2 2 + n 2 3 |  e ε n 2 , e 0 m 2  | 2 |  e aε n 3 , e 0 m 3  | 2 + 1 p n 2 2 + n 2 3  |  e 0 m 3 , e 0 n 2  | 2 − 1   . 20 In the abov e, w e emplo y the analogue of equation (3.10) adapted to the three-dimensional case. No w, pro ceeding similarly as in the pro of of Lemma 4.1 we estimate | ( e ε n 2 , e 0 m 2 ) | 2 | ( e aε n 3 , e 0 m 3 ) | 2 ≤ 8 d 2 d 3 π 4  I n 2 − m 2 + I n 2 + m 2  ·  I n 3 − m 3 + I n 3 + m 3  , (5.14) I n i ± m i = 1 ( n i ± m i ) 2 cos 2  ( n i ± m i ) π (2 t i + ε i ) 2 d i  sin 2  ( n i ± m i ) π ε i 2 d i  . Multiplication of the terms in the brack ets produces fours terms. First fo cus on ∞ X n 2 =1 n 2  = m 2 ∞ X n 3 =1 n 3  = m 3 | G n 2 ,n 3 ( z ) | I n 2 − m 2 I n 3 − m 3 ≤ ∞ X n 2 =1 n 2  = m 2 ∞ X n 3 =1 n 3  = m 3 1 p n 2 2 + n 2 3 sin 2  π ( n 2 − m 2 ) ε d 2  ( n 2 − m 2 ) 2 sin 2  π ( n 3 − m 3 ) aε d 3  ( n 3 − m 3 ) 2 = ∞ X w 2 = − m 2 +1 w 2  =0 ∞ X w 3 = − m 3 +1 w 3  =0 1 p ( w 2 + m 2 ) 2 + ( w 3 + m 3 ) 2 1 w 2 2 w 2 3 sin 2  π w 2 ε d 2  sin 2  π w 3 aε d 3  . Similarly to the proof of Lemma 4.1, we decomp ose the ab o ve sums; ho wev er, this time w e obtain four corresp onding comp onen ts, namely − 1 X − m 2 +1 − 1 X − m 3 +1 , − 1 X − m 2 +1 ∞ X 1 , ∞ X 1 − 1 X − m 3 +1 , ∞ X 1 ∞ X 1 . (5.15) where, in all double sums, the first summation refers to w 2 , and the second to w 3 . In the following, we denote these sums by M 1 , . . . , M 4 , resp ectiv ely . In fact M 1 con tains m 2 comp onen ts, each of which b eha ves as O ( ε 4 ). Therefore, we conclude that M 1 admits O ( ε 4 ) b eha vior. On the other hand, M 2 can b e estimated using the same arguments as in the proof of Lemma 4.1. Indeed the sum P − 1 − m 2 +1 induces the asymptotics O ( ε 2 ) and the sum P ∞ 1 yields the same asymptotics; therefore M 2 = O ( ε 4 ). Due to the symmetry of M 2 and M 3 the same conclusion applies to M 3 . The last sum M 4 in (5.15) requires a more detailed analysis. W e again p erform a decomp osition analogous to (4.10). This yields four terms M 4 , 1 , ..., M 4 , 4 corresp onding to the decomp osition of P w ′ 2 ∈{ ε, 2 ε,...,N 0 } and P w ′ 2 ∈{ N 0 + ε, 2 ε,... } , and analogously for w ′ 3 , cf. (4.10). First we fo cus on M 4 , 1 := ε 6 X w ′ 2 ∈{ ε, 2 ε, 3 ε,...,N 0 } X w ′ 3 ∈{ ε, 2 ε, 3 ε,...,N 0 } 1 p w ′ 2 2 + w ′ 2 3 · 1 w ′ 2 2 w ′ 2 3 · sin 2  w ′ 2 π 2 d 2  · sin 2  aw ′ 3 π 2 d 3  . No w, we estimate the inner sum X w ′ 3 ∈{ ε, 2 ε, 3 ε,...,N 0 } 1 p w ′ 2 2 + w ′ 2 3 · 1 w ′ 2 3 · sin 2  w ′ 3 π 2 d 3  ≤ 1 w ′ 2 · N 0 ε . Applying no w the same reasoning to the external sum with resp ect to w ′ 2 , w e get M 4 , 1 ≤ ε 4 N 2 0 . The further terms, for example corresp onding to P w ′ 2 ∈{ ε, 2 ε,...,N 0 } P w ′ 3 ∈{ N 0 + ε, 2 ε,... } , can b e estimated using the same argumen t and the reasoning of the proof of Lemma 4.1. Summarizing the ab o ve discussion, all sums in (5.15) admit the asymptotics O ( ε 4 ). The remaining expressions of the type I n 2 ± m 2 I n 3 ± m 3 , can b e estimated in the same wa y and shows (5.13). Similarly , an analogue of Lemma 4.2 follows, with obvious modifications. This consequen tly shows that J ε b eha v es as J ε = O ( ε 4 ). This completes the pro of of Theorem 5.1. 21 6 Final remarks and op en questions In the discussed models, the embedded eigenv alues b ecome resonances after in tro ducing a small gap ¯ I ε in the ca vity . The asymptotic behavior of the corresp onding perturbation terms is of order O ( | ¯ I ε | 2 ). This means that, in the t wo-dimensional case, the p erturbativ e comp onen t is of order ε 2 , while in the three-dimensional setting, when the gap has dimensions ε and aε with a > 0, the imaginary component b eha v es as ε 4 . If w e c ho ose the scaling of ¯ I ε suc h that its dimensions are ε and a , then although this case has not b een explicitly considered, relying on the same calculations we obtain resonance asymptotics of order O ( ε 2 ). The imaginary part of z j ( ε ) determines the width of the resonance. The inv erse of ℑ z j ( ε ) characterizes the c haracteristic time scale of the corresponding metastable state and in our case, it b eha ves as τ = O ( | ¯ I ε | − 2 ) . The discussed model, b oth in t w o and three dimensions, admits v arious in teresting generalizations. First, in the three-dimensional setting, the gap can be defined with a more general shap e than a simple rectangle of size [0 , ε ] × [0 , aε ]. It seems reasonable to exp ect that if Ω represents a small gap in a tw o-dimensional baffle with v olume v ol(Ω) tending to zero, then the imaginary parts of the resonance p oles exhibit asymptotics of order O (vol(Ω) 2 ). F urther generalizations concern the shape of the ca vit y , which can b e attached to the w a veguide in v arious w ays — for example, to one of the longitudinal walls. In such configurations, some of the trapped mo des ma y corresp ond to embedded eigenv alues. Ho wev er, in these cases, it is not necessarily true that all of them lie abov e the threshold of the essential sp ectrum. After introducing a small gap in the imp enetrable wall, some of the modes are likely to b ecome discrete eigenv alues located b elo w the contin uum, exhibiting only a small shift in energy . The remaining mo des turn into resonances. How ev er, the models of w av eguides with more complex top ology raise questions whic h, in the context of resonances, remain to b e explored. 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