On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders
aa r X i v : . [ m a t h . SP ] A ug On finitely many resonances emerging under distantperturbations in multi-dimensional cylinders
D.I. Borisov ∗ , A.M. Golovina Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russia,Bashkir State University, Ufa, Russia,University of Hradec Kr´alov´e, Hradec Kr´alov´e, Czech Republic [email protected] Bauman Moscow State Technical University [email protected]
Abstract
We consider a general elliptic operator in an infinite multi-dimensional cylinder with severaldistant perturbations; this operator is obtained by “gluing” several single perturbation operators H ( k ) , k = 1 , . . . , n , at large distances. The coefficients of each operator H ( k ) are periodic in theoutlets of the cylinder; the structure of these periodic parts at different outlets can be different.We consider a point λ ∈ R in the essential spectrum of the operator with several distant per-turbations and assume that this point is not in the essential spectra of middle operators H ( k ) , k = 2 , . . . , n −
1, but is an eigenvalue of at least one of H ( k ) , k = 1 , . . . , n . Under such assumptionwe show that the operator with several distant perturbations possesses finitely many resonancesin the vicinity of λ . We find the leading terms in asymptotics for these resonances, which turnout to be exponentially small. We also conjecture that the made assumption selects the only case,when the distant perturbations produce finitely many resonances in the vicinity of λ . Namely, as λ is in the essential spectrum of at least one of operators H ( k ) , k = 2 , . . . , n −
1, we do expectthat infinitely many resonances emerge in the vicinity of λ .Keywords: distant perturbation, emerging resonance, exponential asymptotics Resonances of unbounded self-adjoint operators are intensively studied both in mathematics andphysics. In mathematics, the notion of resonance is related with an analytic continuation of theresolvent of a considered operator through the essential spectrum and the resonances are defined aspoles of such continuation, see, for instance, monographs [26], [27].Behavior of resonances is studied in various aspects and one of interesting directions in mathematicalphysics is on problems with distant perturbations. A classical example is a Schr¨odinger operator withtwo wells separated by a large distance. A more general case is an elliptic differential operator in anunbounded domain, whose coefficients can be shortly described as several localized profiles connectedby periodic backgrounds, see Figures 1, 2. The behavior of the resolvents and isolated eigenvalues ofsuch operators in various particular cases were studied in a series of works, see, for instance, [1], [3], [4],[12], [13], [17], [18], [21]. General results on the resolvents and isolated eigenvalues of general operatorswith abstract distant perturbations were established in [7], [8], [14], [15], [16].Resonances of the operators with distant perturbations were studied much less. A classical caseof the Schr¨odinger operator with two or several well separated by large distances was considered in[19], [22]. It was found that once the Schr¨odinger operator with one of the wells has a virtual level atthe bottom of its essential spectrum, the multiple well operator has infinitely many resonances nearthe same bottom; the leading terms in the asymptotics for these resonances were found. In [2], aone-dimensional Schr¨odinger operator was considered with a periodic truncated potential, that is, a ∗ Corresponding author | x | > L for L large enough. This is also a model with distantperturbation since such operator can be regarded as glued from two discrete Schr¨dinger operators witha potential being periodic as ± x > ∓ x <
0. A similar discrete model was studiedin [23]. The main result in [2], [23] stated that both discrete and continuous operators with truncatedperiodic potentials had a growing number of closely spaced resonances accumulated along some curvenear the bottom of the essential spectrum. This result is very similar to that of [19], [22] with the onlydifference that the presence of the virtual level was replaced by the truncation of the periodic potential.Very recently, a similar phenomenon was studied in [10], [11] again for a one-dimensional operator withtwo localized perturbations separated by a large distance. The perturbations were differential operatorswith compactly supported coefficients. An important feature was that these perturbations were notsupposed to be symmetric in the operator sense, so, the perturbed operator could be non-self-adjoint.No presence of the virtual level at the bottom of the essential spectrum or the truncation of a periodicpotential were assumed. It was shown that nevertheless, the phenomenon still held. Namely, thereemerges a growing number of resonances or eigenvalues accumulating to a fixed segment in the essentialspectrum. The presence of the eigenvalues is due to non-self-adjointness of the considered operator.The properties of these emerging eigenvalues and resonances were studied in much more details thanin [2], [19], [22], [23]. In particular, these eigenvalues and resonances were found as sums of explicitlywritten convergent series and the error terms in these series were estimated uniformly in an indexcounting these resonances and eigenvalues; these series also served as asymptotic ones. Also a simpleeffective procedure was proposed for finding these eigenvalues and resonances numerically with anarbitrary high precision.Operators with distant perturbations not necessarily always have infinitely many resonances ac-cumulating near some point or a segment in the essential spectrum. They can be situations, whenonly finitely many resonances emerge from a point in the essential spectrum. This was the case in [6],where a Laplacian was considered in the strip with a combination of Dirichlet and Neumann conditionsimposed so that the final model could be regarded as an operator with three distant perturbations andthe central perturbation had a simple isolated discrete eigenvalue λ embedded into the essential spec-trum of the left and right perturbations. It was shown that the original operator had one resonanceconverging to λ and the leading terms in its asymptotics were found.In the present work we consider a general second order scalar differential operator in a multi-dimensional cylinder with finitely many distant perturbations. This operator is obtained via “gluing”several single perturbation operators, each having coefficients periodic outside some compact set, seeFigure 1. We study the resonances of such operator with distant perturbations emerging in the vicinityof a some point λ in its essential spectrum. This point is supposed to be an internal in the essentialspectra of the left or right single perturbation operator, not to belong to the essential spectra of middlesingle perturbations operators and to be an isolated eigenvalue of at least one of the single perturbationoperators. Under this assumption we show that the considered operator with distant perturbationshas finitely many resonances in the vicinity of λ . We find the leading terms of their asymptoticsexpansions; our technique also allows one to construct the next terms in these asymptotics, althoughthis is quite a bulky and technical procedure. The situation we consider is important and deserves anindependent study since it seems to be the only case, when finitely many resonances emerge in thevicinity of the point λ . Once our assumption on λ fails, namely, if λ is in the essential spectrum ofone of middle single perturbation operators, we strongly expect that infinitely many resonances shouldemerge in the vicinity of λ similar to the model studied in [10], [11]. Let x = ( x , x ′ ), x ′ = ( x , . . . , x d ) be Cartesian coordinates in R d and R d − , respectively, d > ω ⊂ R d − be a bounded domain with a boundary in the class C , Π := R × ω be an infinite straightcylinder in R d . By A ( k ) ij = A ( k ) ij ( x ), A ( k ) j = A ( k ) j ( x ), A ( k )0 = A ( k )0 ( x ), i, j = 1 , . . . , d , k = 1 , . . . , n , n > A ( k ) ij , A ( k ) j ∈ W ∞ (Π), A ( k )0 ∈ L ∞ (Π).2he functions A ( k ) ij are assumed to be symmetric and to satisfy an ellipticity condition: A ( k ) ij ( x ) = A ( k ) ji ( x ) , d X i,j =1 A ( k ) ij ( x ) ζ i ζ j > c d X j =1 | ζ j | , x ∈ Π , ζ i ∈ C , where c > x and ζ .We suppose that the functions A ( k ) ♮ , ♮ = i, j ; j ; 0, are periodic in the outlets of the cylinder Π.Namely, there exists a constant a > ± x > a , the functions A ( k ) ♮ are periodic in x , thatis, A ( k ) ♮ ( x − pT ( k − , x ′ ) = A ( k ) ♮ ( x ) , x − a, A ( k ) ♮ ( x + pT ( k ) , x ′ ) = A ( k ) ♮ ( x ) , x > a,♮ = i, j ; j ; 0; i, j = 1 , . . . , d, j = 1 , . . . , d, k = 1 , . . . , n, p ∈ N , where T ( k ) > k = 0 , . . . , n , are some periods. We also assume that A ( k ) ♮ ( a ( k )+ + t, x ′ ) = A ( k +1) ♮ ( − a ( k +1) − + t, x ′ ) , ♮ = i, j ; j ; 0 , x ′ ∈ ω, t T ( k ) , where a ( k ) ± > a are some numbers, i, j = 1 , . . . , d , k = 1 , . . . , n −
1. This condition means that theright periodic part of the function A ( k ) ♮ coincides identically with the left periodic part of the function A ( k +1) ♮ , see a schematic demonstration on Figure 1.By X ( k ) , k = 1 , . . . , n , we denote a set of the numbers obeying the conditions X ( k +1) ℓ − X ( k ) ℓ = a ( k +1) − + a ( k )+ + ( ℓ ( k ) − + ℓ ( k )+ ) T ( k ) , k = 1 , . . . , n − , where ℓ ( k ) ± are some arbitrary natural numbers. Such numbers always exist since it is sufficient tochoose X (1) ℓ and to find other X ( k ) ℓ by the above conditions. Let χ (1) ℓ ( x ) be the characteristic functionof the semi-infinite interval ( −∞ , a (1)+ + ℓ (1)+ T (0) ℓ ], and χ ( k ) ℓ , k = 2 , . . . , n −
1, be the characteristicfunctions of the intervals ( X ( k − − a ( k ) − − ℓ ( k − − T ( k − ℓ , X ( k ) + a ( k )+ + ℓ ( k )+ T ( k ) ℓ ], and finally, χ ( n ) ℓ be thecharacteristic function of the semi-infinite interval [ X ( n ) − a ( n ) − − ℓ ( n ) − T ( n − ℓ , + ∞ ). We denote: A ( ℓ ) ♮ ( x ) := n X k =1 χ ( k ) ℓ ( x ) A ( k ) ♮ ( x − X ( k ) ℓ , x ′ ) . A schematic graph of these functions is provided on Figure 2.The main object of our study is the operator in L (Π) acting as H ℓ = − d X i,j =1 ∂∂x i A ( ℓ ) ij ( x ) ∂∂x j + i d X j =1 (cid:18) A ( ℓ ) j ( x ) ∂∂x j + ∂∂x j A ( ℓ ) j ( x ) (cid:19) + A ( ℓ )0 ( x ) , (2.1)on the domain ˚ W (Π), which is a subspace of the Sobolev space W ( Q ) consisting of the functionswith the zero trace on ∂ Π, while ℓ := ( ℓ ( k ) ± ) k =1 ,...,n − and i is the imaginary unit. The operator H ℓ is self-adjoint and lower-semibounded. We consider the case ℓ ( k ) ± → + ∞ , k = 1 , . . . , n −
1; in whatfollows, we briefly write this fact as ℓ → ∞ .Our main aim is to study the resonances of the operator H ℓ emerging from certain internal pointsin the essential spectrum as ℓ → ∞ , k = 1 , . . . , n −
1. The resonances are introduced as poles of anappropriate local analytic continuation of the resolvent of the operator H ℓ in the vicinity of a point λ ∈ R .Our main assumption on the point λ is formulated in terms of a family of self-adjoint lowersemi-bounded operators in the space L (Π) acting as H ( k ) := − d X i,j =1 ∂∂x i A ( k ) ij ( x ) ∂∂x j + i d X j =1 (cid:18) A ( k ) j ( x ) ∂∂x j + ∂∂x j A ( k ) j ( x ) (cid:19) + A ( k )0 ( x )on the domain ˚ W (Π). This assumption says that λ is an internal point of the essential spectrum of H ℓ , is not in the essential spectra of the operators H ( k ) , k = 2 , . . . , n −
1, but is an eigenvalue of atleast one of the operators H ( k ) , k = 1 , . . . , n . In the next subsection we describe the aforementionedlocal analytic continuation of the resolvent of the operator H ℓ in the vicinity of a point λ ∈ R .3igure 1: Schematic graphs of the coefficients A ( k ) ♮ shown in two copies of the cylinder Π. Figure 2:
Schematic graphs of the coefficients A ( k ) ℓ . In the upper part of the figure two copies of the cylinderΠ are shown and in each of them we provide the sketches of the graphs of the functions A ( k ) ♮ and A ( k +1) ♮ . In thelower part of the cylinder Π we show the graph of the function A ( ℓ ) ♮ glued from the functions A ( k ) ♮ and A ( k +1) ♮ . We first introduce some auxiliary notations and mention known facts. We denote A (0) ♮,per ( x ) := A (1) ♮ ( x ) , x ∈ [ − a (1) − − T (0) , − a (1) − ] × ω,A ( k ) ♮,per ( x ) := A ( k ) ♮ ( x ) , x ∈ [ a ( k )+ , a ( k )+ + T ( k ) ] × ω, k = 1 , . . . , n. We continue the functions A ( k ) ♮,per ( x ) T ( k ) -periodically in x on entire cylinder Π and consider the familyof self-adjoint lower-semibounded periodic operators H ( k ) per := − d X i,j =1 ∂∂x i A ( k ) ij,per ( x ) ∂∂x j + i d X j =1 (cid:18) A ( k ) j,per ( x ) ∂∂x j + ∂∂x j A ( k ) j,per ( x ) (cid:19) + A ( k )0 ,per ( x )in L (Π) on the domain ˚ W (Π). The spectra of these operators have a band structure: σ ( H ( k ) per ) = ∞ [ p =1 n E ( k ) p ( τ ) : τ ∈ (cid:0) − πT ( k ) , πT ( k ) (cid:3) o , where σ ( · ) is the spectrum of an operator, E ( k ) p ( τ ) are the eigenvalues of the corresponding operatoron the periodicity cell (cid:3) ( k ) := (0 , T ( k ) ) × ω taken in the ascending order counting multiplicities.The corresponding operators on the periodicity cells are self-adjoint lower-semibounded operators in L ( (cid:3) ( k ) ) acting asˆ H ( k ) per ( τ ) := d X i,j =1 D i A ( k ) ij,per D j + d X j =1 (cid:0) A ( k ) j,per D j + D j A ( k ) j,per (cid:1) + A ( k )0 ,per , τ ∈ (cid:0) − πT ( k ) , πT ( k ) (cid:3) , (2.2) D := i ∂∂x − τ, D j := i ∂∂x j , j = 2 , . . . , d, on the domains ˚ W ,per ( (cid:3) ( k ) ), which is a subspace in the Sobolev space ˚ W ( (cid:3) ( k ) ) consisting of thefunctions satisfying the periodic boundary conditions on the lateral sides ∂ (cid:3) ( k ) \ ∂ Π of the cell (cid:3) ( k ) and having zero trace on ∂ Π ∩ ∂ (cid:3) ( k ) . 4he following lemma, implied by [9, Lm. 2.1], describes the essential spectrum of the operators H ( k ) and H ℓ ; such spectrum is denoted by the symbol σ e ( · ). Lemma 2.1.
The essential spectra of the operators H ( k ) and H ℓ read as σ e ( H ( k ) ) = σ ( H ( k − per ) ∪ σ ( H ( k ) per ) , k = 1 , . . . , n, σ e ( H ℓ ) = σ ( H (0) per ) ∪ σ ( H ( n ) per ) . We note that the band functions E ( k ) p ( τ ) and the corresponding eigenfunctions Ψ ( k ) p ( x, τ ) can be πT ( k ) -periodically continued in τ on the entire real line preserving their smoothness. In what follows,for the sake of convenience, we assume that this continuation is made on the segment (cid:2) − πT ( k ) , πT ( k ) (cid:3) .Since λ ∈ R is an internal point of the essential spectra of H ℓ , by Lemma 2.1, this means that λ is an internal point in the essential spectrum of at least one of the operators H ( k ) per , k ∈ { , n } .It follows from Theorem 3.9 in [20, Ch. VII, Sect. 5] that there exist points τ ( k ) p ∈ (cid:0) − πT ( k ) , πT ( k ) (cid:3) and numbers δ , δ ( k ) p > H ( k ) per ( τ ) located in the interval[ λ − δ , λ + δ ] form holomorphic in τ ∈ [ τ ( k ) p − δ ( k ) p , τ ( k ) p + δ ( k ) p ] branches ˜ E ( k ) p ( τ ), p = 1 , . . . , M ( k ) ,where M ( k ) are some numbers. The image of each branch ˜ E ( k ) p covers the segment [ λ − δ , λ + δ ].The corresponding eigenfunctions ˜Ψ ( k ) p ( x, τ ) are orthonormalized in L ( (cid:3) ( k ) ) and are holomorphic in τ ∈ [ τ ( k ) p − δ ( k ) p , τ ( k ) p + δ ( k ) p ]. The identities hold:˜ E ( k ) p ( τ ( k ) p ) = λ . (2.3)Our main assumption is the following identities: d ˜ E ( k ) p dτ ( τ ( k ) p ) = 0 , p = 1 , . . . , M ( k ) . (2.4)We stress that we do not suppose the ascending ordering for the eigenvalues ˜ E ( k ) p ( τ ), τ ∈ [ τ ( k ) p − δ ( k ) p , τ ( k ) p + δ ( k ) p ]. The derivative in the left hand side in inequality (2.4) is obviously real.It is clear that the eigenvalues ˜ E ( k ) p ( τ ) and the corresponding eigenfunctions ˜Ψ ( k ) p ( x, τ ) can beholomorphically continued in τ into some neighbourhood of the segment [ τ ( k ) p − δ ( k ) p , τ ( k ) p + δ ( k ) p ] in thecomplex plane; in this sense we regard them as complex functions depending on a complex parameter τ . We fix k , p and in a small neighbourhood of the point λ in the complex plane we consider theequation ˜ E ( k ) p ( τ ) = λ, (2.5)where the parameter τ ranges in the aforementioned neighbourhood of the segment [ τ ( k ) p − δ ( k ) p , τ ( k ) p + δ ( k ) p ] in the complex plane. Conditions (2.3), (2.4) allow us to apply the implicit function theorem toequation (2.5) and to conclude on the unique solvability of this equation in a small neighbourhood ofthe point τ ( k ) p . Without loss of generality we can assume that these neighbourhoods coincide with theaforementioned neighbourhood of the segments [ τ ( k ) p − δ ( k ) p , τ ( k ) p + δ ( k ) p ]. We denote the correspondingsolutions by t ( k ) p ( λ ). These functions are holomorphic in λ in a small neighbourhood of the point λ and satisfy the identities: t ( k ) p ( λ ) = τ ( k ) p . We denote ψ ( k ) p ( x, λ ) := e i t ( k ) p ( λ ) x ˜Ψ ( x ) p ( x, t ( k ) p ( λ )). The symbol ˚ W ,loc (Π) stands for the space of thefunctions in W ,loc (Π) with a zero trace on ∂ Π.The following lemma was proved in [9].
Lemma 2.2.
The functions ψ ( k ) p ( x, λ ) belong to ˚ W ,loc (Π) and solve the equations ( ˆ H ( k ) per − λ ) ψ ( k ) p = 0 in Π . Let d ˜ E ( k ) p dτ ( τ ( k ) p ) > . (2.6) Then the functions ψ ( k ) p ( x, λ ) decay exponentially as x → ±∞ and grow exponentially as x → ∓∞ if ± Im λ > . Let d ˜ E ( k ) p dτ ( τ ( k ) p ) < . (2.7)5 hen the functions ψ ( k ) p ( x, λ ) grow exponentially as x → ±∞ and decay exponentially as x → ∓∞ if ± Im λ > . By P + we denote the subset in the set { , . . . , M ( n ) } of the elements, for which condition (2.6)holds. Similarly, P − is a subset in the set { , . . . , M (0) } of the elements, for which condition (2.7)holds. If λ σ e ( H ( n ) per ), to simplify the notations and calculations, we let P + := ∅ , ψ ( n ) p := 0. If λ σ e ( H (0) per ), we also let P − := ∅ , ψ (0) p := 0.The operators H ( k ) per ( τ ), k ∈ { , . . . , n } , introduced in (2.2) are well-defined also for complex-valued τ . At that, they are no longer self-adjoint for non-real τ but it is still closed on the same domain.We consider the operator H ( k ) per ( τ ) − λ as a quadratic operator pencil with the spectral parameter τ .If τ is an eigenvalue of the operator pencil H ( k ) per ( τ ) − λ , then τ + πT ( k ) is also an eigenvalue and thecorresponding eigenfunctions are related by the multiplying by e π i T ( k ) x . This is in what follows weconsider the eigenvalues of this pencils located only in the strip − πT ( k ) < Re τ πT ( k ) identifying theboundaries of this strip.If λ is a point in the essential spectrum of the operator H ( k ) per , k ∈ { , n } , by Lemma 2.2, thenumbers τ ( k ) p exhaust all real eigenvalues of the operator pencil H ( k ) per ( τ ) − λ ; the correspondingeigenfunctions are ψ ( k ) p ( x, λ ). If λ is not in the essential spectrum of the operator H ( k ) per , the operatorpencil H ( k ) per ( τ ) − λ possesses no real eigenvalues.For an arbitrary b we denote:Π b := Π ∩ { x : | x | < b } , Π + b := Π ∩ { x : x > b } , Π − b := Π ∩ { x : x < b } . Let γ ∈ R , b > ̺ b be the characteristic function of the set Π b . By W , ± ,γ,b we denotethe weighted spaces consisting of the functions u ∈ ˚ W (Π ± b , ∂ Π ± b ∩ ∂ Π) having a finite norm k u k W , ± ,γ,b := k e γx u k W (Π ± b ) . The following theorem describes an analytic continuation of the resolvent of the operator H ℓ ; itwas proved in [9]. Theorem 2.1.
There exists a small fixed neighbourhood Ξ of the point λ in the complex plane inde-pendent of ℓ and the form of the coefficients of the operator H ℓ as − a (0) − + X (1) ℓ < x < a ( n )+ + X ( n ) ℓ suchthat in this neighbourhood, the bordered resolvent ̺ b ( H ℓ − λ ) − ̺ b admits an analytic continuation fromthe upper half-plane into the lower one simultaneously for all b > . The poles of this continuationcalled resonances can be equivalently defined as values of λ , for which there exits a non-trivial solutionin ˚ W ,loc (Π) of the boundary value problem ˆ H ℓ ψ − λψ = 0 in Π , ψ = 0 on ∂ Π ,ψ ( x, λ ) = X p ∈ P + c ± p ψ ± p ( x, λ ) + ˆ ψ ± ( x, λ ) as ± x > ± b ± . (2.8) Here ˆ H ℓ is the differential expression in the right hand side in (2.1), c ± p are some constants, ˆ ψ + ∈ W , +2 ,γ + ,b + and ˆ ψ − ∈ W , − ,γ − ,b − are some functions, b + := X ( n ) ℓ + a , b − := X (1) ℓ − a , and γ + > , γ − < are some numbers chosen so that for all λ ∈ Ξ the strip (cid:8) τ : γ − < Im τ < γ + (cid:9) contains no eigenvaluesof the quadratic operator pencils H (0) per ( τ ) − λ and H ( n ) per ( τ ) − λ except for t ± p ( λ ) . For each k ∈ { , . . . , n } , by N ( k ) we denote the multiplicity of λ considered as an eigenvalue of theoperator H ( k ) and φ ( k ) p = φ ( k ) p ( x ), p = 1 , . . . , N ( k ) are the associated eigenfunctions orthonormalizedin L (Π). If λ is not an eigenvalue of the operator H ( k ) for some k , we let N ( k ) := 0, φ ( k ) p := 0. It isclear that under our assumptions the point λ is a discrete eigenvalue or a point in the resolvent set ofeach operator H ( k ) , k ∈ { , . . . , n − } . We also stress that it can not be an eigenvalue of an infinitemultiplicity for the operators H ( k ) , k ∈ { , n } since such situation is excluded by inequalities (2.4).6n order to formulate our main results, we need to describe the behavior at infinity of the eigen-functions φ ( k ) p associated with the eigenvalue λ . While in simplest cases this issue is trivially resolvedby an appropriate separation of variables, the situation is more complicated for arbitrary periodicallyvarying coefficients. In order to describe this situation, we employ general results from [24]; they pro-vide a needed description in terms of the above introduced quadratic operator pencils H ( k ) per ( τ ) − λ .The final description is not complicated but involves some rather bulky technical details, which wehave to deal with.According [24, Ch. 3, Sect. 4, Prop. 4.4, Stat. 1], the eigenvalues of the quadratic operatorpencils H ( k ) per ( τ ) − λ are isolated, form a countable set and can accumulate at infinity only in an angle | Im τ | > C | Re τ | , C = const . It is easy to see that (cid:0) H ( k ) per ( τ ) − λ (cid:1) ∗ = H ( k ) per ( τ ) − λ , τ ∈ C . The operator H ( k ) per ( τ ) − λ is Fredholm, see [25, Ch. VIII, Sect. 29.3]. Hence, the eigenvalues of theoperator pencil H ( k ) per ( τ ) − λ are complex conjugate. The eigenvalues with positive imaginary parts aredenoted by r ( k, +) i , i = 1 , , . . . , while ones with negative imaginary parts are denoted by r ( k, − ) i = r ( k, +) i , i = 1 , , . . . Taking the multiplicities into account, we arrange these eigenvalues as follows:Im r ( k, +)1 Im r ( k, +)2 Im r ( k, +)3 . . . , Im r ( k, − )1 > Im r ( k, − )2 > Im r ( k, − )3 > . . . We denote r := min k =1 ,...,n − Im r ( k, +)1 . For each k = 1 , . . . , n − H ( k ) per ( τ ) − λ satisfying the conditionIm r ( k, ± ) i = ± r ; (2.9)these eigenvalues are r ( k, ± ) i for i = 1 , . . . , J ( k ) for some J ( k ) >
0. The identity J ( k ) = 0 corresponds tothe situation, when the operator pencil H ( k ) per ( τ ) − λ possesses no eigenvalues r ( k, ± ) i obeying condition(2.9). We note that thanks to an aforementioned discreteness of the set { r k, ± i } i ∈ N , there exists γ ∈ ( r , r ) such that the strip (cid:8) τ : | Im τ | γ (cid:9) contains no other eigenvalues of H ( k ) per ( τ ) − λ except for r ( k, ± ) i , i = 1 , . . . , J ( k ) .For each eigenvalue r ( k, ± ) i there exists an associated Jordan chain Φ ( k, ± ) is , s = 0 , . . . , κ ( k ) i − κ ( k ) i > r ( k, − ) i and r ( k, +) i are same, see [24, Ch. 1, Sect. 2, Prop. 2.2]. The functions Φ ( k, ± ) i are the eigenfunctionsassociated with the eigenvalue r ( k, ± ) i , while the adjoint functions solve the equations (cid:0) H ( k ) per ( r ( k, ± ) i ) − λ (cid:1) Φ ( k, ± ) i + ∂ H ( k ) per ∂τ ( r ( k, ± ) i )Φ ( k, ± ) i = 0 , (cid:0) H ( k ) per ( r ( k, ± ) i ) − λ (cid:1) Φ ( k, ± ) is + ∂ H ( k ) per ∂τ ( r ( k, ± ) i )Φ ( k, ± ) i s − + 12 ∂ H ( k ) per ∂τ ( r ( k, ± ) i )Φ ( k, ± ) i s − = 0 , as s >
2, see [24, Ch. 1, Sect. 2, Subsect. 2].We continue the functions Φ ( k, ± ) is T ( k ) -periodically in x keeping the same notations. We denote ϕ ( k, ± ) is ( x ) := e ± i r ( k, ± ) i x ˜ ϕ ( k, ± ) is ( x ) , ˜ ϕ ( k, ± ) is ( x ) := s X p =0 (i x ) p p ! Φ ( k, ± ) i s − p ( x ) , (2.10)where s = 0 , . . . , κ ( k ) i −
1. According [24, Ch. 5, Sect. 1.3, Thm. 1.4], the introduced functions providethe leading terms in the asymptotics at infinity for the eigenfunctions φ ( k ) p of the operators H ( k ) : φ ( k ) p ( x ) = ˜ φ ( k, ± ) p ( x ) + ˆ φ ( k, ± ) p ( x ) , ± x > a, (2.11)˜ φ ( k, +) p := J ( k ) X i =1 κ ( k ) i − X s =0 α ( k, +) pis ϕ ( k, +) is , k = 1 , . . . , n − , φ ( k, − ) p := J ( k − X i =1 κ ( k − i − X s =0 α ( k − , − ) pis ϕ ( k − , − ) is , k = 2 , . . . , n, where α ( k, ± ) pis are some numbers and ˆ φ ( k, ± ) p ∈ ˚ W , ± , ± γ, ± a are some functions.We also observe that under the shift x x + T , where T is a multiple of the period T ( k ) , thefunction ˜ φ ( k, +) p is transformed as follows:˜ φ ( k, +) p ( x + T, x ′ ) = J ( k ) X i =1 e i r ( k, +) i T κ ( k ) i − X s =0 β ( k, +) pis ( T ) ϕ ( k, +) is ( x ) ,β ( k, +) pis ( T ) := κ ( k ) i − s − X m =0 α ( k, +) pi m + s m ! (i T ) m . (2.12)Similarly, if T is a multiple of T ( k − , then˜ φ ( k, − ) p ( x − T, x ′ ) = J ( k − X i =1 e − i r ( k − , − ) i T κ ( k − i − X s =0 β ( k − , − ) pis ( T ) ϕ ( k − , − ) is ( x ) ,β ( k − , − ) pis ( T ) := κ ( k − i − s − X m =0 α ( k − , − ) pi m + s m ! ( − i T ) m . (2.13)We denote κ := max k =1 ,...,n − i =1 ,...,J ( k ) κ ( k ) i , η ( ℓ ) := k ℓ k κ e − r h ℓ i , h ℓ i := min k =1 ,...,n − (cid:8) | X ( k +1) ℓ − X ( k ) ℓ | (cid:9) , k ℓ k := max k =1 ,...,n − (cid:8) | X ( k +1) ℓ − X ( k ) ℓ | (cid:9) . We let N := n X k =1 N ( k ) . (2.14)and consider an auxiliary block matrix ˚ A = ˚ A ( ℓ ) of size N × N :˚ A = A . . . A A . . . A A . . . . . . A n − n − A n − n . . . A nn − . The block located at the intersection of r th row and k th column is of the size N ( r ) × N ( k ) ; the matrices˚ A (0) k ± k read as˚ A k +1 k ( ℓ ) := (cid:16) ˚A ( k, +) jp ( ℓ ) (cid:17) j =1 ,...,N ( k +1) p =1 ,...,N ( k ) , ˚ A k k +1 ( ℓ ) := (cid:16) ˚A ( k, − ) jp ( ℓ ) (cid:17) j =1 ,...,N ( k ) p =1 ,...,N ( k +1) , (2.15)˚A ( k, +) jp ( ℓ ) := J ( k ) X i =1 e i r ( k, +) i ( X ( k +1) ℓ − X ( k ) ℓ ) J ( k ) X q =1 κ ( k ) i − X s =0 κ ( k ) q − X t =0 α ( k, − ) jqt K ( k ) isqt β ( k, +) pis ( X ( k +1) ℓ − X ( k ) ℓ ) , ˚A ( k, − ) jp ( ℓ ) := J ( k ) X i =1 e − i r ( k, − ) i ( X ( k +1) ℓ − X ( k ) ℓ ) J ( k ) X q =1 κ ( k ) i − X s =0 κ ( k ) q − X t =0 α ( k, +) jqt K ( k ) qtis β ( k, − ) pis ( X ( k +1) ℓ − X ( k ) ℓ ) . · ) j =1 ,...,N ( r ) p =1 ,...,N ( k ) denote the matrices, where the superscript j counts the index of the row,while the subscript p does the index of the column. The constants K ( k ) isqt are defined by the formulae: K ( k ) isqt = 0 as r ( k, +) i = r ( k, +) q , (2.16)and K ( k ) isqt = Z ω Φ ( k, − ) qt ∂ Φ ( k, +) is ∂ν ( k ) − i A ( k )11 ,per Φ ( k, +) i s − ! − Φ ( k, +) is ∂ Φ ( k, − ) qt ∂ν ( k ) + i A ( k )11 ,per Φ ( k, − ) q t − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = T ( k ) dx ′ (2.17)as r ( k, +) i = r ( k, +) q ; as s = 0 or t = 0, in (2.17) we let Φ ( k, +) i − = 0, Φ ( k, − ) q − = 0.By Λ j = Λ j ( ℓ ), j = 1 , . . . , N , we denote the eigenvalues of the matrix ˚ A ( ℓ ) taken counting theiralgebraic multiplities. We partition the set of these eigenvalues into disjoint subgroups { Λ j ( ℓ ) } j ∈ L p , S p L p = { , . . . , N } , so that eigenvalues Λ i and Λ j in the same group satisfy the bound | Λ i ( ℓ ) − Λ j ( ℓ ) | Cη − N ( ℓ ) e − γN h ℓ i , (2.18)while for the eigenvalues in different groups the identity holds: | Λ i ( ℓ ) − Λ j ( ℓ ) | > µ ( ℓ ) η − N ( ℓ ) e − γN h ℓ i . (2.19)The symbol C in (2.18) stands for some fixed constant independent of i , j and ℓ , while the symbol µ ( ℓ ) in (2.19) denotes a positive function such that lim ℓ →∞ µ ( ℓ ) = + ∞ . Theorem 2.2. As ℓ is large enough, there exists a fixed independent of ℓ neighbourhood Ξ of the point λ in the complex plane, in which the operator H ℓ possesses at most N resonances. For each group L p there exists at least one resonance of the operator H ℓ with the asymptotics λ ℓ = λ + Λ i ( ℓ ) + O (cid:0) η − N ( ℓ ) e − γN h ℓ i (cid:1) , ℓ → ∞ , (2.20) where Λ i ∈ L p . The operator H ℓ has at most L p resonances with asymptotics (2.20), where L p isthe number of the elements in the set L p . All eigenvalues Λ i , i = 1 , . . . , N , are of order O ( η ( ℓ )) as ℓ → ∞ and this estimate is order sharp. Let us discuss briefly the main aspects of our problem and of the result. The problem is rathergeneral since the operator H ℓ is multi-dimensional and is “glued” from a fixed but arbitrary number ofdistant perturbations described by the operators H ( k ) . As x ∈ ( X ( k − ℓ , X ( k ) ℓ ), the coefficients of thisoperator are periodic functions A ( k ) ♮,per , which are not supposed to be same for different k . Our modelis quite rich and includes, in particular, various situations like truncated periodic potentials similarto one considered in [23], multiple potential and magnetic wells separated by large distances, distantperturbations of metric, etc. Our main assumption is formulated in the end of Subsection 2.1 in termsof the spectra of the operators H ( k ) ; additional assumption (2.4) is rather technical and is made forsimplicity. Under these assumptions, our main result states that the operator H ℓ can have only finitelymany resonances in the vicinity of λ and all these resonances converge to λ and have asymptotics(2.20). The total number of different resonances is at most N but can be less. Of course, if all Λ i aredistinct, we have exactly N different resonances. But in the general situation some resonances of theoperator H ℓ can coincide. If this is the case, the total multiplicity of all perturbed resonances as thenumber of associated linear independent solutions of problem (2.8) is again at most N but can be lesssince instead of some such generalized eigenfunctions, adjoint functions can exist. We also stress thatif λ is not an eigenvalue of all operators H ( k ) as k = 1 , . . . , n , then our result implies that the operator H ℓ has no resonances in the vicinity of λ .As we have said above, we assume that λ / ∈ σ e ( H ( k ) ), k = 2 , . . . , n −
1, and thanks to Lemma 2.1,this is equivalent to λ / ∈ σ e ( H ( k ) per ), k = 2 , . . . , n −
1. This assumption is very essential since it ensuresthat the operator H ℓ has only finitely many resonances in the vicinity of the point λ . If λ is in theessential spectrum of at least one of the operators H ( k ) per , k = 2 , . . . , n −
1, then the situation changes9ramatically. As we have said in Introduction, particular examples considered in [2], [10], [11], [19]show that in this case there can be a growing number of closely spaced resonances in the vicinityof the point λ . We do expect that such situation always occurs in our general model once λ is inthe essential spectrum of at least one of the operators H ( k ) per , k = 1 , . . . , n −
1. This is an interestingproblem, which we shall study in our next work. Right now we stress that in view of our conjecture,the situation studied in the present work is the only case, when finitely many resonances emerge fromthe point λ .It should be also said that the leading terms Λ i in asymptotics (2.20) are exponentially small since η is exponentially small. Hence, all considered resonances of the operator H ℓ are exponentially closeto λ . Although we provide only leading terms in asymptotics (2.20), our technique allows one to findnext terms in this asymptotics. The point is that all resonances of the operator H ℓ are zeroes of certainequation, see (4.26). In fact, in the proof of Theorem 2.2, we find a leading term for the matrix A inthis equation, which turns out to be ˚ A and its eigenvalues determines the leading terms in (2.20). Italso possible to find a more detailed asymptotics for the matrix A and this will determine next termsin asymptotics (2.20); these are rather bulky and technical but simple calculations. We also mentionthat at least formally, next terms in asymptotics (2.20) can be constructed via the scheme proposedin [16, Sect. 5]; however, an issue of justification of this scheme for the resonances seems to be morecomplicated than for the isolated eigenvalues studied in [16]. In the present section we collect some preliminaries on the behavior of solutions to some auxiliaryproblems, which will be then employed in the proof of our main result. We introduce two auxiliaryspaces W := W (Ω − ) ⊕ W (Ω + ) , Ω ± := { x ∈ Π : 0 < ± x < a } , V := L (Π a ) ⊕ W ( ω ) ⊕ W ( ω ) , and an operator family F ( k ) ( λ ) : W → V acting by the following rule. Given g ∈ W , the componentsof the vector F ( k ) ( λ ) g = ( F, f , f ) ∈ V are defined as F := ( ˆ H ( k ) − λ ) g (cid:12)(cid:12)(cid:12) Ω ± on Ω ± , , f := [ g ] , f := (cid:20) ∂g∂ν ( k ) (cid:21) . (3.1)We consider boundary value problems: (cid:0) ˆ H ( k ) − λ (cid:1) u ( k ) = F in Π \ ω , u = 0 on ∂ Π , [ u ( k ) ] = f , (cid:20) ∂u ( k ) ∂ν ( k ) (cid:21) = f , (3.2) u ( k ) ∈ W (Π +0 ) as k = 1 , . . . , n − , u ( n ) ∈ W ,loc (Π +0 ) ,u ( k ) ∈ W (Π − ) as k = 2 , . . . , n, u (1) ∈ W ,loc (Π − ) , (3.3) u (1) ( x, λ ) = X p ∈ P − (cid:0) c p, − ( λ ) g (cid:1) ψ − p ( x, λ ) + ˆ u (1) − ( x, λ ) as x < − a, ˆ u (1) − ( x, λ ) ∈ W , − , ˜ r ,a , (3.4) u ( n ) ( x, λ ) = X p ∈ P + (cid:0) c p, + ( λ ) g (cid:1) ψ + p ( x, λ ) + ˆ u (1)+ ( x, λ ) as x > a, ˆ u ( n )+ ( x, λ ) ∈ W , +2 , ˜ r ,a , (3.5)where [ v ] := v (cid:12)(cid:12) x =+0 − v (cid:12)(cid:12) x = − for an arbitrary v ∈ W (cid:0) Π a \ ω ) (cid:1) , ω := { } × ω, ∂∂ν ( k ) := − d X i =1 A ( k )1 i,per ∂∂x i + i A ( k )1 ,per ,c p, ± ( λ ) are some linear bounded functionals on V and 0 < ˜ r < r is some fixed number. By T ± : L (Π) → L (Ω ± ) we denote the operator of restriction on Ω ± , that is, T ± u := u (cid:12)(cid:12) Ω ± .A following lemma describes the behavior of solutions of problems (3.2), (3.3) as k ∈ { , . . . , n − } .10 emma 3.1. Let λ be a discrete eigenvalue of the operator H ( k ) , k ∈ { , . . . , n − } . There exists asmall neighbourhood Ξ k of the point λ such that for λ ∈ Ξ k \ { λ } problems (3.2), (3.3) are uniquelysolvable and the following statements hold. The operators U ( k ) = U ( k ) ( λ ) : W → W (Π − ) ⊕ W (Π +0 ) (3.6) mapping g into the solution of these problems can be represented as U ( k ) = N ( k ) X p =1 φ ( k ) p λ − λ P ( k ) p ( λ ) + R ( k ) ( λ ) , (3.7) P ( k ) p ( λ ) g := ( F, φ ( k ) p ) L (Π a ) + ( f , φ ( k ) p ) L ( ω ) − (cid:18) f , ∂φ ( k ) p ∂ν ( k ) (cid:19) L ( ω ) , (3.8) where R ( k ) ( λ ) are the reduced resolvents being bounded operators from W into W (Π − ) ⊕ W (Π +0 ) holomorphic in λ ∈ Ξ k . The reduced resolvents act into the orthogonal complement in L (Π) to theeigenspace spanned over the eigenfunctions φ ( p ) k , p = 1 , . . . , N ( k ) , and solve boundary value problem(3.2), (3.3) with F replaced by F − N ( k ) P p =1 φ ( k ) p P ( k ) p g .For λ ∈ Ξ k and positive sufficiently large X , the estimates hold: kT ± S ( ± X ) U ( k ) ( λ ) k W → W (Ω ± ) Ce − r X , (3.9) where c and C are some constant independent of = w , X and λ . The operators T S ( ± X ) U ( k ) ( λ ) areholomorphic in λ ∈ Ξ k .Proof. We seek a solution to problem (3.2), (3.3) as u ( k ) = v + χg , where ξ = ξ ( x ) is an infinitelydifferentiable cut-off function equalling to one as | x | < a and vanishing as | x | > a . Then for a newunknown function v we obtain the equation( H ( k ) − λ ) v = ˜ F, ˜ F := ( H ( k ) − λ )(1 − ξ ) g. (3.10)According formula (3.21) and other results in [20, Ch. 5, Sect 3.5], this equation is uniquely solvablefor λ ∈ Ξ k \ { } , where Ξ k is some sufficiently small fixed neighbourhood of the point λ in the complexplane. The solution is meromorphic in λ having a simple pole at λ : v = ( H ( k ) − λ ) − = N ( k ) X p =1 ( ˜ F, φ ( k ) p ) L (Π a ) λ − λ φ ( k ) p + ˜ R ( λ ) ˜ F, (3.11)where ˜ R ( λ ) is a reduced resolvent; this is a bounded operator from L (Π a ) into ˚ W (Π) holomorphic in λ ∈ Ξ k . The reduced resolvent acts in the orthogonal complement in L (Π a ) to the eigenspace spannedover the eigenfunctions φ ( p ) k , p = 1 , . . . , N ( k ) . It solves the equation( H ( k ) − λ ) ˜ R ( λ ) ˜ F = ˜ F − N ( k ) X p =1 ( ˜ F, φ ( k ) p ) L (Π a ) φ ( k ) p . (3.12)Let us find the scalar products ( ˜ F, φ ( k ) p ) L (Π a ) . Employing the definition of the function ˜ F and theeigenvalue equation for the functions φ ( k ) p , we integrate by parts:( ˜ F, φ ( k ) p ) L (Π a ) = Z Π a φ ( k ) p ( ˜ H ( k ) − λ )(1 − ξ ) g dx = Z Π a φ ( k ) p F dx − Z Π a \ ω φ ( k ) p ( ˜ H ( k ) − λ ) ξ g dx + ( λ − λ ) Z Π a φ ( k ) p ξ g dx = P ( k ) p ( λ ) g + ( λ − λ )( ξ g, φ ( k ) p ) L (Π a ) . (3.13)11e substitute this formula into (3.11) and then for u ( k ) we obtain representation (3.7) with R ( λ ) g = ˜ R ( λ ) ˜ F + ξ g − N ( k ) X p =1 ( ξ g, φ ( k ) p ) L (Π a ) φ ( k ) p . (3.14)Hence, in view of the aforementioned properties of the operator ˜ R ( λ ), the operator R ( λ ) is boundedfrom W into W (Π − ) ⊕ W (Π +0 ) and is holomorphic in λ ∈ Ξ k . A desired boundary value problem for R ( λ ) g follow identity (3.14) and problem (3.12).We proceed to studying the operator T S ( X ) U ( k ) ( λ ); the case of the operator T S ( − X ) U ( k ) ( λ ) canbe studied in the same way. Let ξ = ξ ( x ) be an infinitely differentiable cut-off function equallingto one as x > a + 2 and vanishing as x < a + 1. As x > a , the coefficients of the operator H ( k ) areperiodic and hence, in view of the problem for R ( λ ) g , the function w = ξ R ( λ ) g solves the equation( H ( k ) per − λ ) w = ˘ F − N ( k ) X p =1 ξ φ ( k ) p P ( k ) p g, (3.15)where ˘ F = ˘ F ( x, λ ) is some compactly supported function in L (Π) being non-zero only as x ∈ ( a + 1 , a + 2); the mapping g ˘ F is linear bounded operator from W into L (Π) holomorphic in λ ∈ Ξ k .Since ( H ( k ) per ( τ ) − λ ) − = (cid:0) I + ( λ − λ )( H ( k ) per ( τ ) − λ ) − (cid:1) − (cid:0) H ( k ) per ( τ ) − λ (cid:1) − , where I is the identity mapping, and the inverse operator (cid:0) H ( k ) per ( τ ) − λ (cid:1) − has poles at the eigenvalues r ( k ) i of order at most κ , [24, Ch. 1, Sect. 2, Thm. 2.8], it is easy to confirm that the eigenvalues ofthe operator pencil H ( k ) per ( τ ) − λ are located in C | λ − λ | κ -neighbourhoods of the eigenvalues r ( k, ± ) i ,where C is some fixed constant. Hence, lessening the neighbourhood Ξ k if it is needed, we concludethat the operator pencil H ( k ) per ( τ ) − λ has no eigenvalues in the strip | Im τ | < r as λ ∈ Ξ k . Then wecan apply Theorem 1.4 in [24, Ch. 5, Sect. 1.3] to problem (3.15) and we conclude that w ∈ and themapping g
7→ T S ( X ) w is a linear bounded operator from W into W (Π a ) holomorphic in λ ∈ Ξ k andsatisfying estimate (3.9). Since u = w as x > a + 2, this implies all desired properties of the operator T S ( X ) U ( k ) . The proof is complete. Remark 3.1.
We observe that if λ is not in the spectrum of the operator H ( k ) , then the statementof Lemma 3.1 also holds with N ( k ) = 0 and ( H ( k ) − λ ) − = R ( k ) ( λ ) . We recall that by our assumptions, the point λ can be an eigenvalue of the operators H ( k ) , k ∈{ , n } , of multiplicity N ( k ) , embedded into their essential spectra; the identity N ( k ) = 0 correspondsto the case, when λ is not an eigenvalue of H ( k ) . The next lemma describes the behavior of solutionsof problems (3.2), (3.3), (3.4), (3.5) as k ∈ { , n } . Lemma 3.2.
There exists small fixed neighbourhoods Ξ k , k ∈ { , n } , such that problems (3.2), (3.3),(3.4), (3.5) are uniquely solvable respectively as λ ∈ Ξ \ { λ } and λ ∈ Ξ n \ { λ } and the followingstatements hold. The operators (3.6) satisfy representation (3.7) with functionals (3.8), where R ( k ) ( λ ) are the reduced resolvents being bounded operators from W into W ,loc (Π − ) ⊕ W (Π +0 ) for k = 1 and into W (Π − ) ⊕ W ,loc (Π +0 ) for k = n ; these operators are holomorphic in λ ∈ Ξ k . The reducedresolvents satisfy the orthogonality conditions Z Π φ ( k ) p R ( k ) ( λ ) g dx = 0 , p = 1 , . . . , N ( k ) . The functions R ( k ) ( λ ) g solve problems (3.2), (3.3), (3.4), (3.5) with F replaced by F − N ( k ) P p =1 φ ( k ) p P ( k ) p g .For λ ∈ Ξ k and positive sufficiently large X , the estimates hold: kT S ( − X ) R (1) ( λ ) k W → W (Π a ) Ce − r X , kT S ( X ) R ( n ) ( λ ) k W → W (Π a ) Ce − r X , (3.16)12 here c and C are some constant independent of = w , X and λ and constant c satisfies the secondinequality in (3.9). The operators T S ( − X ) U (1) ( λ ) and T S ( X ) U ( n ) ( λ ) are holomorphic in λ ∈ Ξ k , k ∈ { , n } .Proof. We prove the lemma only for k = 1; the case k = n can be treated in the same way. As inthe proof of Lemma 3.1, we seek a solution as u (1) = ξ g + v and for a new unknown function v weobtain problem (3.2), (3.3), (3.4) with F replaced by ˜ F defined in (3.10). According [9, Thm. 1.3], theobtained problem for v is uniquely solvable and the solution satisfies representation (3.14) for v withthe only difference that now the operator ˜ R ( λ ) acts into W ,loc (Π − ) ⊕ W (Π +0 ) and is holomorphic in λ ∈ Ξ in this sense. The rest of the proof reproduces that of Lemma 3.1. In the present section we begin the proof of our main result, Theorem 2.2. The study of the resonancesof the operator H ℓ emerging in the vicinity of the point λ is reduced first to a special operator equationand then we show that these resonances coincide with the zeroes of certain holomorphic function. Ourscheme is based on approach suggested in work [6], but there are serious modification due to a muchmore general formulation of the problem.Throughout the proof we fix a small neighbourhood Ξ of the point λ such that Ξ ⊂ Ξ k for all k ∈ { , n } and Ξ is a subset of the neighbourhood mentioned in Theorem 2.1. In what follows weassume that λ ∈ Ξ and if needed, this neighbourhood will be lessen without saying this explicitly.
In this subsection we show that all non-trivial solutions of problem (2.8) associated with resonances λ ∈ Ξ satisfy certain representation, which will serve as a base for the proof of Theorem 2.2.We choose arbitrary g k ∈ W , k = 1 , . . . , n , assuming that g ≡ − and g n ≡ + . By u k = u k ( x, λ ) we denote the solutions of problems (3.2), (3.3), (3.4), (3.5) with g = g k . Let S ( · ) be ashift operator in L (Π) acting by the rule S (cid:0) · (cid:1)(cid:0) u ( x ) (cid:1) := u ( x − · , x ′ ).The main aim of the present subsection is to prove the following lemma. Lemma 4.1.
For ℓ large enough each non-trivial solution of problem (2.8) associated with some λ ∈ Ξ can be represented as ψ ( x, ℓ ) = u ( x − X (1) ℓ , x ′ ) as x < X (1) ℓ ,u k − ( x − X ( k − ℓ , x ′ )+ u k ( x − X ( k ) ℓ , x ′ ) as X ( k − ℓ < x < X ( k ) ℓ , k = 2 , . . . , n,u n ( x − X ( n ) ℓ , x ′ ) as x > X ( n ) ℓ . (4.1) for some g k ∈ W . The rest of the subsection is devoted to the proof of this lemma. Let ψ be a non-trivial solution ofproblem (2.8) associated with some λ ∈ Ξ and assume that ℓ is large enough. In view of formula (4.1),the restriction of the function u on Π − and that of u n on Π +0 are recovered immediately: u ( x ) := ψ ( x − X (1) ℓ , x ′ ) , x < , u n ( x ) := ψ ( x − X ( n ) ℓ , x ′ ) , x > . (4.2)We are going to show how to recover u ( x ) as x > u ( x ) as x <
0; the procedure for otherfunctions is similar.We choose some f ± ∈ W (Ω ± ) and consider two boundary value problems in W (Π − ) ⊕ W (Π +0 ): (cid:0) ˆ H (1) per − λ (cid:1) v ± = 0 in Π \ ω , v ± = 0 on ∂ Π , [ v ± ] = ± f ± (cid:12)(cid:12) x = ± , (cid:20) ∂v ± ∂ν (1) (cid:21) = ± ∂f ± ∂ν (1) (cid:12)(cid:12)(cid:12)(cid:12) x = ± . (4.3)13ince λ / ∈ σ ( H ( k ) per ), k = 1 ,
2, a statement similar to Lemma 3.1 holds for problem (4.3). Namely,these problems are uniquely solvable. The operators U ( k ) per ( λ ), k = 1 ,
2, mapping f ± into the solutionsare holomorphic in λ ∈ Ξ as bounded operators from W (Ω ± ) into W (Π − ) ⊕ W (Π +0 ). Moreover, for X large enough the estimates hold: kT − S ( − X ) U (1) per ( λ ) k W → W (Ω − ) Ce − cX , kT + S ( X ) U (2) per ( λ ) k W → W (Ω + ) Ce − cX , (4.4)where C and c are some fixed positive constants independent of λ ∈ Ξ.We first recover the function u ( x ) as x > a and u ( x ) as x < − a . In other words, we recoverthe functions u ( x − X (1) ℓ , x ′ ) and u ( x − X (2) ℓ , x ′ ) as X (1) ℓ + a < x < X (2) ℓ − a . In this zone, thecoefficients of the operator H ℓ are pure periodic and ˆ H ℓ coincides with ˆ H (1) per , the needed functions u and u should solve the equation ( ˆ H (1) per − λ ) u k = 0, k = 1 ,
2, should vanish on the ∂ Π and they shouldalso satisfy the boundary conditions( u ( x − X (1) ℓ , x ′ ) + u ( x − X (2) ℓ , x ′ )) (cid:12)(cid:12) x = X (1) ℓ + a +0 = ψ ( x ) (cid:12)(cid:12) x = X (1) ℓ + a +0 , ( u ( x − X (1) ℓ , x ′ ) + u ( x − X (2) ℓ , x ′ )) (cid:12)(cid:12) x = X (2) ℓ − a − = ψ ( x ) (cid:12)(cid:12) x = X (2) ℓ − a − , and the same condition for the first derivatives ∂∂ν (1) . The function u should decay as x grows, while u should decay as x goes in the opposite direction.The main idea how to find u and u is a follows. We first treat u as an unknown function,and then u = ψ − u . We write the discussed equation and boundary conditions for u at x = X (1) ℓ + a + 0 as appropriate boundary value problem (4.3) for v + = u . The values of the function ψ ( x − X ℓ (1) − a ) − v + ( x ) at x = X (2) ℓ − X (1) ℓ − a serve as the boundary conditions for the function u = v − in corresponding boundary value problem (4.3). Once we know the function u , we considerits values at x = − ( X (2) ℓ − X (1) ℓ − a ) and we recall that these values were initially considered as anunknown function. So, we arrive at certain functional equation for u , which turns out to be uniquelysolvable. This determines uniquely both u and u . We proceed to a rigorous realization of this idea.We choose an arbitrary f ∈ W (Ω + ) and let v + := U (1) per ( λ ) S ( − X (1) ℓ − a ) ψ − U (1) per ( λ ) f. (4.5)In terms of this function, we define one more function v − as v − := U (2) per ( λ ) T − S ( − X (2) ℓ ) ψ − U (2) per ( λ ) T − S ( − X (2) ℓ − X ℓ + 2 a ) v = U (2) per ( λ ) T − S ( − X (2) ℓ ) ψ − U (2) per ( λ ) T − S ( − X (2) ℓ − X ℓ + 2 a ) U (1) per ( λ ) S ( − X (1) ℓ − a ) ψ + U (2) per ( λ ) T − S ( − X (2) ℓ − X ℓ + 2 a ) U (1) per ( λ ) f. (4.6)Then we postulate that f = T + S ( X (2) ℓ − X (1) ℓ − a ) v − (4.7)and in view of (4.6) this gives rise to an equation: f − L ( λ, X ) f = h, X := X (2) ℓ − X (1) ℓ − a, (4.8)where h := T + S ( X ) U (2) per ( λ ) T − S ( − X ) ψ − T + S ( X ) U (2) per ( λ ) T − S ( − X ) U (1) per ( λ ) S ( − X (1) ℓ − a ) ψ, L ( λ, X ) := T + S ( X ) U (2) per ( λ ) T − S ( − X ) U (1) per ( λ ) . Thanks to estimates (4.4), we see immediately that the operator L ( λ, X ) is bounded in the space W (Ω + ) and its norm is exponentially small in X : kL ( λ, X ) k W (Ω + ) → W (Ω + ) Ce − cX . f and the functions v ± viaformulae (4.5), (4.6). In view of the latter formulae, it is straightforward to check that the function v ( x ) := v + ( x ) as x < ,v + ( x ) + v − ( x − X, x ′ ) as 0 < x < X,v − ( x − X, x ′ ) as x > X, solves the boundary value problem (cid:0) ˆ H (1) per − λ (cid:1) v = 0 in Π \ ( ω ∪ ω X ) , v = 0 on ∂ Π , [ v ] = S ( − X (1) ℓ − a ) ψ (cid:12)(cid:12) x =+0 , (cid:20) ∂v∂ν (1) (cid:21) = ∂∂ν (1) S ( − X (1) ℓ − a ) ψ (cid:12)(cid:12) x =+0 , [ v ] X = − S ( − X (1) ℓ − a ) ψ (cid:12)(cid:12) x = X − , (cid:20) ∂v∂ν (1) (cid:21) X = − ∂∂ν (1) S ( − X (1) ℓ − a ) ψ (cid:12)(cid:12) x = X − , (4.9)where ω X := { X } × ω, [ v ] X := v (cid:12)(cid:12) x = X +0 − v (cid:12)(cid:12) x = X − . This problem is uniquely solvable, which can be proved as the same was done for problems (3.2), (3.3)in Lemma 3.1. It is easy to see that the function x x < ,ψ ( x + X (1) ℓ + a, x ′ ) as 0 < x < X, x > X, solves problem (4.9). Hence, ψ ( x + X (1) ℓ + a, x ′ ) = v ( x ) and therefore, v + ( x ) + v − ( x − X, x ′ ) = ψ ( x + X (1) ℓ + a, x ′ ) as x ∈ (0 , X ) , (4.10) v + ( x ) = 0 as x < , v − ( x ) = 0 as x > . (4.11)Then we let u ( x ) := v + ( x − a, x ′ ) as x > a, u ( x ) := v − ( x + a, x ′ ) as x < − a. (4.12)It remains to find the function u for x ∈ (0 , a ) and the function u for x ∈ ( − a, u ( x ) := ψ ( x − X (1) ℓ , x ′ ) − u ( x − X (2) ℓ + X (1) ℓ , x ′ ) as x ∈ (0 , a ) ,u ( x ) := ψ ( x − X (2) ℓ , x ′ ) − u ( x + X (2) ℓ − X (1) ℓ , x ′ ) as x ∈ ( − a, , (4.13)We observe that in the above definition of the function u , respectively, u , we employ the values ofthe function u , respectively, u , already defined in (4.12).It follows from (4.11), (4.7), (4.5), (4.6) that the function u defined in (4.12), (4.13) satisfies theidentities: u ( a + 0 , x ′ ) − u ( a − , x ′ ) = v + (+0 , x ′ ) − ψ (cid:0) a − X (1) ℓ , x ′ (cid:1) + u (cid:0) a − X (2) ℓ + X (1) ℓ , x ′ (cid:1) = v + ( − , x ′ ) = 0 . In the same way we confirm that ∂u ∂ν (1) ( a + 0 , x ′ ) = ∂u ∂ν (1) ( a − , x ′ ) ,u ( − a + 0 , x ′ ) = u ( − a − , x ′ ) , ∂u ∂ν (1) ( − a + 0 , x ′ ) = ∂u ∂ν (1) ( − a − , x ′ ) . Hence, the found function u belong to W (Π − ) ⊕ W (Π +0 ), while the function u belongs to W (Π − ).It also follows from (4.10) and (4.2) that u (cid:0) x − X (1) ℓ , x ′ (cid:1) + u (cid:0) x − X (2) ℓ , x ′ (cid:1) = ψ ( x ) as x < X (2) ℓ . The function u also solves problem (3.2), (3.3), (3.4), (3.5) with g = (0 ⊕ T + S ( X (2) ℓ − X (1) ℓ ) u ). Italso determines partially g , namely, T − g = T − S ( X (1) ℓ − X (2) ℓ ) u . Other functions u j and g j can berecovered by repeating the above described procedure for x ∈ ( X ( k ) ℓ , X ( k +1) ℓ ), k = 2 , . . . , n −
1. Thiscompletes the proof of Lemma 4.1. 15 .2 Reduction to operator equation
Here we reduce problem (2.8) to some operator equation, which is more convenient for further purposes.We introduce a Hilbert space G := ( g = g . . .g n , g k ∈ W , k = 1 , . . . , n, g (cid:12)(cid:12) Ω − = 0 , g n (cid:12)(cid:12) Ω + = 0 ) , ( g , h ) G = n X k =1 ( g k , h k ) W . Then we choose an element g ∈ G and construct a function ψ by formula (4.1). We know by Lemma 4.1that all nontrivial solutions of problem (2.8) associated with resonances λ ∈ Ξ are of form (4.1), so,instead of finding ψ , we are going to find a corresponding g ∈ G .The function ψ introduced by (4.1) satisfies boundary conditions in (2.8) and possesses a neededbehavior at infinity. We only need to confirm that it belongs to W ,loc (Π) and solves the equationin (2.8). The former condition is ensured by the continuity in the trace sense of ψ and ∂ψ∂ν ( k ) at { X ( k ) ℓ } × ω , k = 1 , . . . , n . The equation is to be checked only as | x − X ( k ) ℓ | < a since outside thesezones the equation is obviously satisfied. In view of the definition of the functions u k and formulae(3.1) it is easy to see that both the belonging to W ,loc (Π) and the validity of the equation hold once g = − (cid:0) ⊕ T + u ( · + X (1) ℓ − X (2) ℓ ) (cid:1) ,g k = − (cid:0) T − u k − ( · + X ( k ) ℓ − X ( k − ℓ ) ⊕ T − u k +1 ( · + X ( k ) ℓ − X ( k +1) ℓ ) (cid:1) , k = 2 , . . . , n − ,g n = − (cid:0) T − u n − ( · + X ( n ) ℓ − X ( n − ℓ ) ⊕ (cid:1) . These identities can be rewritten as a system of operator equations g + T g = 0 ,g k + T kk − g k − + T kk +1 g k +1 = 0 , k = 2 , . . . , n − ,g n + T nn − g n − = 0 , (4.14)where T kj are operators in the space W defined as T k k +1 ( λ, ℓ ) g k +1 = T + S ( X ( k +1) ℓ − X ( k ) ℓ ) u k +1 , T k k − ( λ, ℓ ) g k − = T − S ( X ( k ) ℓ − X ( k − ℓ ) u k − , In the space G we introduce an operator: T ( λ, ℓ ) := T . . . T T . . . T T . . . . . . T n − n − T n − n −
00 0 0 0 . . . T n − n − T n − n . . . T nn − . In terms of the above notation, system (4.14) casts into the form: g + T ( λ, ℓ ) g = 0 . (4.15)The sought resonances of the operator H ℓ are the value of λ ∈ Ξ, for which the latter equation possessesnon-trivial solutions. The further study of this equation is based on the approach suggested in [7], [8],see also [5]. This approach will allow us to reduce the above problem on nontrivial solutions of (4.15)to searching zeroes of some holomorphic function.
We recall that λ is an eigenvalue of some of the operators H ( k ) , k ∈ { , . . . , n } , of multiplicities N ( k ) with associated orthonormalized in L (Π) eigenfunctions φ ( k ) p , p = 1 , . . . , N ( k ) ; the identity N ( k ) = 016orresponds to the case, when λ is not an eigenvalue of the operator H ( k ) . We denote Φ ( k ) p, + ( ℓ ) := T − S (cid:16) X ( k ) ℓ − X ( k +1) ℓ (cid:17) φ ( k ) p ∈ G , p = 1 , . . . , N ( k ) , where the non-zero element stands as ( k + 1)th position and Φ ( k ) p, − ( ℓ ) := T + S (cid:16) X ( k ) ℓ − X ( k − ℓ (cid:17) φ ( k ) p ∈ G , p = 1 , . . . , N ( k ) , where the non-zero element stands as ( k − N ( k ) = 0, the above vectors are supposedto be zero. We also denote Φ (1) p := Φ (1) p, + , Φ ( n ) p := Φ ( n ) p, − , Φ ( k ) p := Φ ( k ) p, − + Φ ( k ) p, + . By (2.10), (2.11), (2.12), (2.13), the estimate holds: k Φ ( k ) p k G C k ℓ k κ e − r h ℓ i , (4.16)where C > ℓ .In view of Lemmata 3.1, 3.2 and the definition of the functions u k , for λ close to λ , the operator T ( λ, ℓ ) admits the representation: T ( λ, l ) g = 1 λ − λ n X k =1 N ( k ) X p =1 C ( k ) p ( λ ) g Φ ( k ) p ( ℓ ) + R ( λ, ℓ ) g . (4.17)Here C ( k ) p = C ( k ) p ( λ ) are the functionals on G defined as C ( k ) p ( λ ) g := P ( k ) p ( λ ) g k , (4.18)and the writing C ( k ) p ( λ ) g Φ ( k ) p ( λ, ℓ ) denotes a usual scalar multiplication of the number C ( k ) p ( λ ) g andthe vector Φ ( k ) p ( λ, ℓ ). The symbol R = R ( λ, ℓ ) stands for the following operator in G : R := R . . . R R . . . R R . . . . . . R n − n − R n − n −
00 0 0 0 . . . R n − n − R n − n . . . R nn − , where R kj = R kj ( λ, ℓ ) are the operators in W defined by the formulae R k k +1 := T + S ( X ( k +1) ℓ − X ( k ) ℓ ) R ( k +1) ( λ ) , R k k − := T + S ( X ( k − ℓ − X ( k ) ℓ ) R ( k − ( λ ) , R ( k ± from representations (3.7). In view of estimates (3.9), (3.16), the operators R kj acting in L (Π a +1 ) possess an exponentially small norm and the same is true for the operator R in the space G . Namely, there exist C > c > λ ∈ Ξ such that for all λ ∈ Ξ theestimates hold kR k − k ( λ, ℓ ) k C | X ( k ) l − X ( k − l | κ e − ( r − c | Im λ | ) k ℓ k , kR k +1 k ( λ, ℓ ) k C | X ( k +1) l − X ( k ) l | κ e − ( r − c | Im λ | ) k ℓ k , kR ( λ, ℓ ) k C k ℓ k κ e − ( r − c | Im λ | ) k ℓ k (4.19)for sufficiently large h ℓ i . Moreover, the operator R ( λ, ℓ ) is holomorphic in λ ∈ Ξ. In particular, thismeans that there exists an inverse operator Q = Q ( λ, ℓ ) := (cid:0) I + R ( λ, ℓ ) (cid:1) − (4.20)holomorphic in λ , where I is the identity mapping.We substitute representation (4.17) into equation (4.15) and apply then the operator Q : g + 1 λ − λ n X k =1 N ( k ) X p =1 C ( k ) p ( λ ) g Q ( λ, ℓ ) Φ ( k ) p ( ℓ ) = 0 . (4.21)We apply the functionals C ( r ) j ( λ ) to the obtained equation: C ( r ) j ( λ ) g + 1 λ − λ n X k =1 N ( k ) X p =1 C ( k ) p ( λ ) g C ( r ) j ( λ ) Q ( λ, ℓ ) Φ ( k ) p ( ℓ ) = 0 . (4.22)These equations are a system of linear algebraic equations with respect to unknown quantities C ( r ) j ( λ ) g .We observe first that only nontrivial solutions of system (4.22) can generate nontrivial solution ofproblem (2.8). Indeed, by equation (4.21), the trivial solution C ( k ) p ( λ ) g = 0 gives g = 0. In its turn,this means that the corresponding functions u k are also trivial and the same is true for the function ψ defined by formula (4.1).To study the existence of nontrivial solutions to system (4.22), we rewrite it first to a matrix form.We introduce the vector of unknowns in a block form: C := C ... C n , C k = C k ( λ ) := (cid:16) C ( k ) p ( λ ) g (cid:17) p =1 ,...,N ( k ) Hereinafter the vectors of the form ( · ) p =1 ,...,N ( k ) are treated as the vector columns. If N ( k ) = 0, thecorresponding column in the above formulae is absent. The total size of the introduced column is equalto N defined by (2.14).We introduce a matrix A = A ( λ, ℓ ) of size N × N . This matrix has a block form: A := A . . . A n ... ... A n . . . A nn . (4.23)Each block A rk = A rk ( λ, ℓ ) is of the size N ( r ) × N ( k ) . The blocks are defined as A rk ( λ, ℓ ) := (cid:16) C ( r ) j ( λ ) Q ( λ, ℓ ) Φ ( k ) p (cid:17) j =1 ,...,N ( r ) p =1 ,...,N ( k ) . (4.24)In view of the introduced notations, equations (4.22) are rewritten to the matrix one: (cid:18) E + 1 λ − λ A ( λ, ℓ ) (cid:19) C = 0 . (4.25)18y the Cramer’s rule, the existence of nontrivial solution is equivalent to the equationdet (cid:18)(cid:0) λ − λ (cid:1) E − A ( λ, ℓ ) (cid:19) = 0 , (4.26)where E is the unit matrix. This identity is the equation for the sought resonances of the operator H ℓ . For each root λ = λ ( ℓ ) of this equation, the corresponding nontrivial solution of system (4.25)generates the solution g of equation (4.21). In its turn, by formula (4.1), this solution generates asolution to problem (2.8). Thus, we need to study the existence and behaviour of roots of equation(4.26). This will be done in the next section. In this section we complete the proof of Theorem 2.2. Our strategy is as follows. First we establish apreliminary rough estimate for the matrix A , which allows us to prove the solvability of equation (4.26)and to localize the roots, that is, to prove that all roots are contained in a circle of an exponentiallysmall radius centered at λ . The next step is devoted to describing the asymptotics of the matrix A as ℓ → ∞ . Employing this asymptotics, in a final step we find leading terms in the asymptotics of theroots.Throughout this section, by C we denote various inessential constants independent of sufficientlylarge ℓ and λ ∈ Ξ. By B r ( z ) we denote the ball in the complex plane of a radius r centered at a point z . We begin with an obvious property implied by Lemmata 3.1, 3.2, namely, a holomorphic dependenceof the matrix A in λ ∈ Ξ. This implies that the function in the left hand side in (4.26) is holomorphicin λ .The third estimate in (4.19) and definition (4.20) of the operator Q yield immediately that thisoperator is bounded uniformly in λ ∈ Ξ and sufficiently large h ℓ i . Then inequality (4.16) and definition(4.23), (4.24) of the matrix A yield the following bound for the matrix A : k A ( λ, ℓ ) k Cη ( ℓ ) . (5.1)Here as a norm for the matrix A , we choose the maximal among the absolute values of its entries.We calculate the determinant in the left hand side in equation (4.26) rewriting this equation as( λ − λ ) N + F( λ, ℓ ) = 0 , F( λ, ℓ ) := N − X i =0 F i ( λ, ℓ )( λ − λ ) i , (5.2)where F i ( λ, ℓ ) are some functions holomorphic in λ ∈ Ξ; thanks to (5.1), these functions obey theestimates | F i ( λ, ℓ ) | Cη ( N − i ) ( ℓ ) . (5.3)It follows from (5.2), (5.3) that each root λ ∈ Ξ of equation (4.26) satisfies the estimate | λ − λ | Cη N , (5.4)and hence, it converges to λ as ℓ → ∞ . Then we consider the circle B δ ( λ ) ⊂ Ξ of a fixed radius δ and by (5.3) we see that | F i ( λ, ℓ ) | < | λ − λ | N = δ n as λ ∈ ∂B δ ( λ ) . This estimate and the aforementioned holomorphy of F in λ ∈ Ξ allow us to apply the Rouch´e theoremand to conclude that the function in the left hand side in equation (4.26) possesses exactly the sameamount of the zeroes in B δ ( λ ), counting their orders, as the function λ ( λ − λ ) N does. Hence,equation (4.26) has exactly N roots in Ξ counting their orders; by (5.4), all these roots tend to λ as ℓ → ∞ . 19inally, we are going to improve estimate (5.4). We consider the circle B cη ( ℓ ) ( λ ), where c := 2+2 C ,where C is from (5.3). By (5.3), on the boundary of this circle we have the estimate: | F( λ, ℓ ) | N − X i =0 | F i ( λ, ℓ ) | c N − i η N − i ( ℓ ) c N − C − c − η N ( ℓ ) Cc N − η N ( ℓ ) < c N η N ( ℓ ) = | λ − λ | N . Hence, we can apply the Rouch´e theorem once again and we see that equation (4.26) has exactly N roots, counting their orders, in the circle B cη ( ℓ ) ( λ ). This means that all these roots satisfy theestimate: | λ − λ | cη ( ℓ ) . This is the desired estimate for the roots of equation (4.26). A In the present subsection we find an asymptotics for the matrix A as ℓ → ∞ . Since all roots of equation(4.26) are located in the circle B cη ( ℓ ) ( λ ), in what follows we consider only λ ∈ B cη ( ℓ ) ( λ ).Estimate (4.19) and definition (4.20) of the operator Q allow us to expand the latter operator intothe standard Neumann series, which implies, in particular, the representations: Q ( λ, ℓ ) = I − R ( λ, ℓ ) Q ( λ, ℓ ) , kR ( λ, ℓ ) Q ( λ, ℓ ) k Cη ( ℓ ) , where the operator R ( λ, ℓ ) Q ( λ, ℓ ) is holomorphic in λ ∈ B cη ( ℓ ) ( λ ). We also observe that since λ ∈ B cη ( ℓ ) ( λ ), it follows from the definition of the functionals C ( k ) j that (cid:13)(cid:13) C ( k ) j ( λ ) − C ( k ) j ( λ ) (cid:13)(cid:13) Cη ( ℓ ) . Hence, in view of (4.16) and by the definition of the matrix A , we infer that it satisfies the representation A ( λ, ℓ ) = B ( ℓ ) + A ( λ, ℓ ) , (5.5)where B := B . . . B n ... ... B n . . . B nn , B rk ( ℓ ) := (cid:16) C ( r ) j ( λ ) Φ ( k ) p (cid:17) j =1 ,...,N ( r ) p =1 ,...,N ( k ) , and A is some matrix holomorphic in λ ∈ B cη ( ℓ ) ( λ ) obeying the estimate: k A ( λ, ℓ ) k Cη ( ℓ ) . (5.6)According [24, Ch. 5, Sect. 1.3], the functions ϕ ( k, ± ) is introduced in (2.10) solve boundary valueproblems ( ˆ H ( k ) per − λ ) ϕ ( k, ± ) is = 0 in Π , ϕ ( k, ± ) is = 0 on ∂ Π . (5.7)We first prove an auxiliary lemma. Lemma 5.1.
The identities K ( k ) isqt = ± lim N →±∞ Z ω ϕ ( k, − ) qt ∂ϕ ( k, +) is ∂ν ( k ) − ϕ ( k, +) is ∂ϕ ( k, − ) qt ∂ν ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = NT ( k ) dx ′ , (5.8) hold, where the limits in the right hand sides are finite and are independent of the choice of the signin their definition.Proof. We choose an arbitrary natural m large enough and integrate twice by parts in the followingintegral:0 = Z Π mT ( k ) ϕ ( k, − ) qt (cid:0) ˆ H ( k ) per − λ (cid:1) ϕ ( k, +) is dx = Z ω ϕ ( k, − ) qt ∂ϕ ( k, +) is ∂ν ( k ) − ϕ ( k, +) is ∂ϕ ( k, − ) qt ∂ν ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = mT ( k ) dx ′ Z ω ϕ ( k, − ) qt ∂ϕ ( k, +) is ∂ν ( k ) − ϕ ( k, +) is ∂ϕ ( k, − ) qt ∂ν ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = − mT ( k ) dx ′ . This implies that if the limits in the right hand side in (5.8) exist, then they are independent of thechoice of the sign.Let ξ = ξ ( x ) be an infinitely differentiable cut-off function equalling to one as x > x
1. By problems (5.7), the integrand in the integral Z Π mT ( k ) ϕ ( k, − ) qt (cid:0) ˆ H ( k ) per − λ (cid:1) χ ϕ ( k, +) is dx is compactly supported and this is why the integral is independent on m large enough. Integrating byparts and bearing in mind problems (5.7) and definition (2.11) of the functions ϕ ( k, ± ) is , for sufficientlylarge m we obtain: Z Π mT ( k ) ϕ ( k, − ) qt (cid:0) ˆ H ( k ) per − λ (cid:1) ξ ϕ ( k, +) is dx = Z ω ϕ ( k, − ) qt ∂ϕ ( k, +) is ∂ν ( k ) − ϕ ( k, +) is ∂ϕ ( k, − ) qt ∂ν ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = mT ( k ) dx ′ = e i m (cid:0) r ( k, +) i − r ( k, − ) q (cid:1) T ( k ) Z ω ˜ ϕ ( k, − ) qt ∂ ˜ ϕ ( k, +) is ∂ν ( k ) − ˜ ϕ ( k, +) is ∂ ˜ ϕ ( k, − ) qt ∂ν ( k ) + i (cid:0) r ( k, +) i − r ( k, − ) q (cid:1) ˜ ϕ ( k, +) is ˜ ϕ ( k, − ) qt ∂x ∂ν ( k ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = mT ( k ) dx ′ . (5.9)By definition (2.11) of the functions ˜ ϕ ( k, ± ) is and the periodicity of the functions Φ ( k, ± ) is , the latterintegral in (5.9) depends polynomially on m . We also recall that r ( k, − ) q = r ( k, +) q .Let r ( k, +) i = r ( k, +) q . Then the exponent e m i( r ( k, +) i − r ( k, − ) q ) T ( k ) = e i m ( r ( k, +) i − r ( k, +) q ) T ( k ) oscillates in m ;at that, it can increases or decreases in m . In this case, identities (5.9) are possible for all sufficientlylarge m only as all integrals in these identities vanish. This proves formulae (2.16).Let r ( k, +) i = r ( k, +) q . In this case the exponent in the latter integral in (5.9) disappears. The remainingintegral is polynomial in m and identities (5.9) are possible only as this polynomial degenerates into itsfree coefficient. It is easy confirm that this coefficient is exactly the right hand side in formula (2.17).The proof is complete.All components of the vector Φ ( k ) p ( ℓ ) are zero except those at ( k − k + 1)th positions.Hence, by definition (4.18) of the functionals C ( r ) j we see immediately that B rk ( ℓ ) ≡ | r − k | 6 = 1 . (5.10) Lemma 5.2.
The matrices A k ± k satisfy the representations A k ± k ( λ , ℓ ) = ˚ A k ± k ( ℓ ) + O (cid:0) e − γ h ℓ i (cid:1) , ℓ → ∞ , (5.11) where, we recall, the matrices ˚ A k ± k ( ℓ ) were defined in (2.15).Proof. The entries of the matrices ˚ A k ± k are determined by the quantities C ( k ± j ( λ ) Φ ( k ) p ( ℓ ). Let usfind out their asymptotic behavior. By formulae (2.11), (2.12) we obtain: T − S ( X ( k ) ℓ − X ( k +1) ℓ ) φ ( k ) p = J ( k ) X i =1 e i r ( k, +) i ( X ( k +1) ℓ − X ( k ) ℓ ) κ ( k ) i − X s =0 β ( k, +) pis ( X ( k +1) ℓ − X ( k ) ℓ ) T − ϕ ( k, +) is + O (cid:0) e − γ | X ( k +1) ℓ − X ( k ) ℓ | (cid:1) .
21n the same way we find T + S ( X ( k +1) ℓ − X ( k ) ℓ ) φ ( k +1) p = J ( k ) X i =1 e i r ( k, − ) i ( X ( k +1) ℓ − X ( k ) ℓ ) κ ( k ) i − X s =0 β ( k, − ) pis ( X ( k +1) ℓ − X ( k ) ℓ ) T + ϕ ( k, − ) is + O (cid:0) e − γ | X ( k +1) ℓ − X ( k ) ℓ | (cid:1) . It follows from the obtained formulae, identities (2.10), definition (4.18) of the functional C ( r ) j , andLemma 5.1 that C ( k +1) j ( λ ) Φ ( k ) p, + ( ℓ ) = J ( k ) X i =1 e i r ( k, +) i ( X ( k +1) ℓ − X ( k ) ℓ ) κ ( k ) i − X s =0 β ( k, +) pis ( X ( k +1) ℓ − X ( k ) ℓ ) · P ( k +1) j F ( k +1) ( λ )( T + ϕ ( k, +) is ⊕
0) + O (cid:0) e − γ | X ( k +1) ℓ − X ( k ) ℓ | (cid:1) , (5.12)According (3.8), the quantity P ( k +1) p F ( k +1) ( λ )( T + ϕ ( k, +) is ⊕
0) is given by formula P ( k +1) j ( λ )( T + ϕ ( k, +) is ⊕
0) = Z Ω − φ ( k +1) j ( H ( k +1) − λ ) ϕ ( k, +) is dx + Z ω φ ( k +1) j ∂ϕ ( k, +) is ∂ν ( k +1) − ϕ ( k, +) is ∂φ ( k +1) p ∂ν ( k +1) ds. As x < − a , thanks to the definition of the function ϕ ( k, +) is , the equation holds( H ( k +1) − λ ) ϕ ( k, +) is = ( H ( k +1) per − λ ) ϕ ( k, +) is = 0and this is why we can integrate parts as in (5.9): P ( k +1) j ( λ )( T + ϕ ( k, +) is ⊕
0) = lim m →−∞ Z ( − mT ( k ) , × ω φ ( k +1) j ( H ( k +1) − λ ) ϕ ( k, +) is dx + Z ω φ ( k +1) j ∂ϕ ( k, +) is ∂ν ( k +1) − ϕ ( k, +) is ∂φ ( k +1) j ∂ν ( k +1) ds = lim m →−∞ Z ω ϕ ( k, +) is ∂φ ( k +1) j ∂ν − φ ( k +1) j ∂ϕ ( k, +) is ∂ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = mT ( k ) dx ′ = J ( k ) X q =1 κ ( k ) q − X t =0 α ( k, − ) jqt lim m →−∞ Z ω ϕ ( k, +) is ∂ϕ ( k, − ) qt ∂ν − ϕ ( k, − ) qt ∂ϕ ( k, +) is ∂ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = mT ( k ) dx ′ = J ( k ) X q =1 κ ( k ) q − X t =0 α ( k, − ) jqt K ( k ) isqt . Substituting these identities into relations (5.12), we arrive at a final formula: C ( k +1) j ( λ ) Φ ( k ) p, + ( ℓ ) = ˚A ( k, +) jp ( ℓ ) + O (cid:0) e − γ | X ( k +1) ℓ − X ( k ) ℓ | (cid:1) . In the same way we confirm that P ( k ) j ( λ )(0 ⊕ T − ϕ ( k, − ) is ) = J ( k ) X q =1 κ ( k ) q − X t =0 α ( k, +) jqt K ( k ) qtis , and C ( k ) j ( λ ) Φ ( k +1) p, − ( ℓ ) = ˚A ( k, − ) jp ( ℓ ) + O (cid:0) e − γ | X ( k +1) ℓ − X ( k ) ℓ | (cid:1) . The obtained formulae yield (5.11), (2.15). The proof is complete.22 .3 Asymptotics for the roots
In this subsection we find the asymptotics for the roots of equation (4.26). We begin with observingthat identity (5.10) and Lemma 5.2 imply B ( ℓ ) = ˚ A ( ℓ ) + O (cid:0) e − γ h ℓ i (cid:1) . Then by (5.5), (5.6) we infer that A ( λ, ℓ ) = ˚ A ( ℓ ) + A ( λ, ℓ ) , (5.13)where A ( λ, ℓ ) is a holomorphic in λ ∈ B cη ( ℓ ) ( λ ) matrix obeying the estimate k A ( λ, ℓ ) k Ce − γ h ℓ i . Having this estimate and identity (5.13) in mind as well as the fact that Λ j are the eigenvalues of thematrix ˚ A , we rewrite equation (4.26) as N Y j =1 (cid:0) λ − Λ j ( ℓ ) (cid:1) + G ( λ, ℓ ) = 0 , where G is a holomorphic in λ ∈ B cη ( ℓ ) ( λ ) function obeying the estimate | G ( λ, ℓ ) | Ce − γ h ℓ i η N − ( ℓ ); (5.14)the factor η N − appears in the latter estimate since | λ − λ | < cη ( ℓ ).It follows from Lemma 5.2 that the matrix ˚ A is of order O ( η ) and this estimate is order sharp.Hence, the same is true for its eigenvalues.We choose a group L p and take one of the eigenvalues Λ j in this group. We consider the ball B cϑ (Λ j ), where ˜ c > ϑ ( ℓ ) := η − N ( ℓ ) e − γN h ℓ i . Weassume that the constant ˜ c is such that { Λ i } i ∈ L p ⊂ B cϑ (Λ j ) and the distances from Λ j to theboundary ∂B cϑ (Λ j ) is at least ˜ cϑ . Hence, Y i ∈ L p | λ − Λ i | > (˜ cϑ ) | L p | as λ ∈ ∂B cϑ (Λ j ) . (5.15)It follows from (2.19) that for Λ i / ∈ L p a similar estimate holds: Y i/ ∈ L p | λ − Λ i | > (cid:18) µϑ (cid:19) N −| L p | as λ ∈ ∂B cϑ (Λ j ) . (5.16)This inequality and (5.15) imply: Y i ∈{ ,...,N } | λ − Λ i | > ˜ cϑ N = ˜ ce − γ h ℓ i η N − ( ℓ ) as λ ∈ ∂B cϑ (Λ j ) . In view of this estimate and (5.14) we see that choosing ˜ c = 2 C , we get (cid:12)(cid:12)(cid:12)(cid:12) Y i ∈{ ,...,N } ( λ − Λ i ) (cid:12)(cid:12)(cid:12)(cid:12) > | G ( λ, ℓ ) | as λ ∈ ∂B cϑ (Λ j )and by Rouch´e theorem, equation (4.26) has exactly the same number of zeroes in B cϑ (Λ j ) as thefunction λ Q i ∈{ ,...,N } ( λ − Λ i ). The zeroes of the latter function in B cϑ (Λ j ) are exactly Λ i , i ∈ L p .Hence, equation (4.26) has the same number of roots counting their orders in B cϑ (Λ j ). All these rootssatisfy | λ ( ℓ ) − Λ j ( ℓ ) | < cϑ and this proves asymptotics (2.20). In a general situation, λ ( ℓ ) is complex-valued and in this case its imaginary part is negative, since otherwise the corresponding non-trivialsolution to problem (2.8) would be an eigenfunction associated with a complex-valued eigenvalue, whatis impossible. If the root λ ( ℓ ) is real, then the associated non-trivial solution to problem (2.8) can bean eigenfunction of the operator H ℓ but this situation still fits our definition of the resonance. Theproof of Theorem 2.2 is complete. 23 cknowledgements The authors thank A.A. Fedotov and S.A. Nazarov for useful comments and discussion some aspectsof the work.The results presented in Sections 4.2, 4.3, 5 were financially supported by Russian Science Founda-tion (grant no. 17-11-01004).
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