Extreme localisation of eigenfunctions to one-dimensional high-contrast periodic problems with a defect

Following a number of recent studies of resolvent and spectral convergence of non-uniformly elliptic families of differential operators describing the behaviour of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a"defect": an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period $\varepsilon$ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues $\lambda_\varepsilon$ that are induced by the defect and converge to a point $\l_0$ located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are"extremely"localised to the defect, in the sense that the localisation exponent behaves as $\exp(-\nu/\varepsilon),$ $\nu>0,$ which has not been observed in the existing literature. As a consequence, we argue that $\l_0$ is an eigenvalue of a certain limit operator defined on the defect only. In two- and three-dimensional configurations, whose one-dimensional cross-sections are described by the setting considered, this implies the existence of propagating waves that are localised to the defect. We also show that the unperturbed operators are norm-resolvent close to a degenerate operator on the real axis, which is described explicitly.

1. Introduction. The question of whether a macroscopic perturbation of material properties in a periodic medium or structure (periodic composite) induces the existence of a localized solution (bound state) to the time-harmonic version of the equations of motion is of special importance from the physics, engineering, and mathematical points of view. Depending on the application context, such a solution can have either an advantageous or undesirable effect on the behavior of systems containing the related composite medium as a component. For example, in the context of photonic (phononic) crystal fibers, perturbations of this kind have been exploited for the transport of electromagnetic (elastic) energy over large distances with little loss into the surrounding space; see, e.g., [14], [17]. In the mathematics literature, proofs of the existence or nonexistence of such a localized solution have been carried out using the tools of the classical asymptotic analysis of the governing equations and spectral analysis of operators generated by the governing equations in various natural"" function spaces. The choice of the concrete class of equations and functions under study is usually motivated by the applications in mind, and several works that have marked the development of the related analytical techniques cover a wide range of operators and their relatively compact perturbations, e.g., [20], [3], [2], [10]. The present work is a study of localization properties for a class of composite media that has been the subject of increasing interest in the mathematics and physics literature recently, in view of its relation to the so-called metamaterials, e.g., manufactured composites possessing negative refraction properties. It has been shown in [8] that the spectrum of a stratified high-contrast composite, represented mathematically by a one-dimensional periodic second-order differential equation, has an infinitely increasing number of gaps (lacunae) opening in the spectrum, in the limit of the small ratio \varepsi between the period and the overall size of the composite. This analytical feature, analogous to the spectral property of multidimensional high-contrast periodic composites shown in [22], provides a mathematical recipe for the use of such materials in the physics context or technologies where the presence of localized modes (generated by defects in the medium) has important practical implications. In the physical context of photonic crystal fibers and within the mathematical setup of multidimensional high-contrast media, this link has been studied in [12], [5], [6]. In the paper [12], a two-scale asymptotics for eigenfunctions of a high-contrast second-order elliptic differential operator with a finite-size perturbation (defect) was derived. It was shown that for eigenvalues \lambda in gaps of the spectrum of the (two-scale) operator representing the leading-order term of this asymptotics, there are sequences of eigenvalues of the finite-period problems that converge to \lambda as \varepsi \rightar 0. The subsequent works [5], [6] developed a multiscale version of Agmon's approach [1] and proved that the corresponding eigenfunctions of the limit operator decay exponentially fast away from the defect. An important technical assumption in all these works is that the low-modulus inclusions in the composite have a positive distance to the boundary of the period cell, which is not possible to satisfy in one dimension.
In the more recent paper [8], a family of nonuniformly elliptic periodic onedimensional problems with high contrast was studied, which in practically relevant situations corresponds to a stratified composite with alternating layers of homogeneous media with highly contrasting material properties. It was shown that the spectra of the corresponding operators converge, as \varepsi \rightar 0, to the band-gap spectrum of a two-scale operator described explicitly in terms of the original material parameters. Introducing a finite-size defect D into the setup of [8], one is led to consider the operator -\bigl( a \varepsi D u \prime \bigr) \prime , a \varepsi D > 0, where a \varepsi D takes values of order one on D and is \varepsi -periodic (\varepsi > 0) in \BbbR \setminu D with alternating values of order one and \varepsi 2 . As was mentioned in [8, section 5.1], a formal analysis suggests that the rate of decay of eigenfunctions localized in the vicinity of the perturbation D is``accelerated exponential,"" rather than just exponential as in [6], in the sense that the decay exponent increases in absolute value as \varepsi \rightar 0. The goal of the present work is to provide a rigorous proof of this property, formulated below as Theorem 2.4. In view of the above discussion, this new localization property can be seen as a consequence of two features of the underlying periodic composite: loss of uniform ellipticity (via the presence of soft inclusions in a moderately stiff material) and the one-dimensional nature of the problem.
In addition to our main result, we formulate (section 3) a new characterization of the limit spectrum for the unperturbed family of problems in the whole space discussed in [8] and strengthen (section 6) the result of [8] by proving an order-sharp norm-resolvent convergence estimate for this family (Theorem 2.2). In particular, this new estimate implies order-sharp uniform asymptotic estimates, as \varepsi \rightar 0, for the related family of evolution semigroups; cf., e.g. [23] for a strong-convergence version of this kind of result. for a j , a - 1 j , \rho j , \rho - 1 j \in L \infty (Y j ), j = 0, 1, periodic with period 1. It is convenient to set a 0 \equiv 0 on Y 1 and a 1 \equiv 0 on Y 0 ; thus we can write, for example, a \varepsi (x) = \varepsi 2 a 0 (x/\varepsi ) + a 1 (x/\varepsi ). We denote \Omega 0 := \bigcup z\in \BbbZ (Y 0 + z), \Omega 1 := \bigcup z\in \BbbZ (Y 1 + z) and reserve the notation z for an integer, unless stated otherwise. We will refer to the sets \Omega \varepsi 0 , \Omega 0 and \Omega \varepsi 1 , \Omega 1 as the soft and stiff component, respectively. For a positive Lebesgue-measurable function w on a Borel set B \subset \BbbR , such that w, w - 1 \in L \infty (B), we employ the notation L 2 w (B) to indicate that the space L 2 (B) is equipped with the inner product For a closed and semibounded sesquilinear form \beta : H 1 (\BbbR ) \times H 1 (\BbbR ) \rightar \BbbC , the (selfadjoint) operator A associated to \beta is densely defined in L 2 w (\BbbR ) by the action Au = f, where for a given f \in L 2 w (\BbbR ), the function u \in H 1 (\BbbR ) is the solution to the integral identity Henceforth, all function spaces we employ consist of complex-valued functions and are over \BbbC .
For the sesquilinear form we consider A \varepsi , the operator defined in L 2 \rho \varepsi (\BbbR ) and associated to \beta \varepsi . The spectrum \sigma (A \varepsi ) of A \varepsi is absolutely continuous and, by introducing the rescaled Floquet--Bloch transform \scrU \varepsi (see (6.1)), we note that \sigma (A \varepsi ) admits the representation where \sigma (A \varepsi \theta ) is the spectrum of the L 2 \rho (Y ) densely defined self-adjoint operator A \varepsi \theta associated to the form acting in the space H 1 \theta (Y ) of functions u \in H 1 (Y ) that are \theta -quasiperiodic, i.e., such that u(y) = exp(i\theta y)v(y), y \in Y, for some 1-periodic function v \in H 1 (Y ). For each \varepsi , \theta , the operator A \varepsi \theta has compact resolvent and consequently its spectrum \sigma (A \varepsi \theta ) is discrete.
Consider the space and its closure in L 2 \rho (Y ), which we denote by V \theta , which we also equip with the norm of L 2 \rho (Y ). We introduce the densely defined operators A \theta in V \theta given by A \theta u = f, where for all f \in V \theta the function u in the domain of A \theta is such that For each \theta , the operator A \theta has compact resolvent, and so \sigma (A \theta ) is discrete. In a recent work [8] (see also section 6 of the present manuscript), the spectrum \sigma (A \varepsi ) was shown to converge in the Hausdorff sense to the union of the spectra of the operators A \theta , i.e., lim \varepsi \rightar0 \bigcup \theta \in [0,2\pi ) \sigma (A \theta ) can be seen as the spectrum of a certain operator A 0 unitary equivalent to the direct integral of operators \int \oplus A \theta ; see Appendix A for the details.
In section 6, we construct infinite-order asymptotics (as \varepsi \rightar 0) for the resolvents of A \varepsi \theta , uniform in \theta , with respect to the H 1 -norm and, in particular, prove the following refinement of the result established in [8].
Here, \{ \lambda \varepsi n (\theta )\} n\in \BbbN , \{ \lambda n (\theta )\} n\in \BbbN are the eigenvalue sequences of A \varepsi \theta , A \theta , respectively, labeled in the increasing order. 1 It follows that for sufficiently small \varepsi , the spectrum \sigma (A \varepsi ) has a gap if the set \bigcup \theta \sigma (A \theta ) contains a gap. In section 3 we give an example of a class of coefficients for which this set contains infinitely many gaps. Furthermore, we demonstrate that \lambda \in \bigcup \theta \sigma (A \theta ) if and only if the inequality \bigm| \bigm| \bigm| Here v 1 and v 2 are the (\lambda -dependent) solutions of subject to the conditions Remark 2.3. Note that any solution u of - (a 0 u \prime ) \prime = \lambda \rho 0 u is absolutely continuous and so is its coderivative a 0 u \prime . Hence, their value at any point y is well defined (unlike the value of a 0 or u \prime in general). This explains the use of notation (a 0 v \prime j )(y), which we will hold to throughout the paper.
Next, we introduce d -, d + \in \BbbR , and on the set D = (d -, d + ) replace the coefficients (2.1) by some uniformly positive and bounded functions a D , \rho D ; namely, we consider We shall study the spectrum of the operator A \varepsi D defined in L 2 \rho \varepsi D (\BbbR ) and associated to the form As this operator arises from a compact perturbation of the coefficients of A \varepsi , it is well-known that the essential spectra of A \varepsi D and A \varepsi coincide; see, e.g., [10]. For eigenvalues situated, for small values of \varepsi , in the gaps of the essential spectrum of A \varepsi D (equivalently, in the gaps of the essential spectrum of A \varepsi ), we expect the eigenfunctions to be localized around the defect. In view of the above observation about the spectra of A \varepsi and A \theta , \theta \in [0, 2\pi ), we are therefore interested in the analysis of eigenfunctions of A \varepsi D corresponding to eigenvalues that are located in the gaps of the limit spectrum \bigcup \theta \sigma (A \theta ). Consider the operator A N,D defined in L 2 \rho D (D) and associated to the form acting in H 1 (D). The functions from the domain of A N,D satisfy the Neumann condition on the boundary of D. We show that if the defect D is chosen so that the spectrum \sigma (A N,D ) has a nonempty intersection with \BbbR \setminu \bigcup \theta \sigma (A \theta ), then for sufficiently small \varepsi the operator A \varepsi D has nonempty point spectrum. Notice that we can always choose a D , \rho D , dand d + such that this is true. Moreover, we demonstrate that for eigenvalue sequences that converge to a point in \BbbR \setminu \bigcup \theta \sigma (A \theta ) the corresponding eigenfunctions are localized to a small neighborhood of the defect. Namely, the eigenfunctions u \varepsi exhibit accelerated exponential decay outside the defect in the sense that the function exp \bigl( dist(x, D)\nu /\varepsi \bigr) u \varepsi (x), x \in \BbbR , is an element of L 2 (\BbbR \setminu D) for sufficiently small \varepsi , where the value \nu is determined by the distance of the limit point of \lambda \varepsi to the set \bigcup \theta \sigma (A \theta ). These results are collated in the following theorem, which we prove in sections 4 and 5. 1. For every \lambda 0 \in \sigma (A N,D )\setminu \bigcup \theta \sigma (A \theta ) (which is always simple) there exists a sequence of simple eigenvalues \lambda \varepsi of A \varepsi D converging to \lambda 0 and constants C 1 , C 2 > 0 such that where u 0 is a normalized eigenfunction of A N,D corresponding to the eigenvalue \lambda 0 , the set Then, for sufficiently small values of \varepsi , the function g \nu /\varepsi u \varepsi is an element of L 2 (\BbbR ) for all \nu < \bigm| \bigm| ln | \mu 1 | \bigm| \bigm| . Remark 2.5. One can improve the eigenvalues convergence rate at least to | \lambda \varepsi -\lambda 0 | \leq \widetil C\varepsi and in a rather generic case even to | \lambda \varepsi -\lambda 0 | \leq \widetil C\varepsi 2 for some \widetil C > 0 (improving accordingly the convergence estimate for the eigenfunctions), by``attaching"" the periodic structure to the defect D in a``correct"" way; see the end of section 4 and Theorem 4.2 for the details.  By taking test functions v \in C \infty 0 (Y 0 ) we deduce that u| Y0 is a weak solution to the equation -(a 0 u \prime ) \prime = \lambda \rho 0 u on Y 0 . For L \infty -functions a 0 and \rho 0 , (3.2) holds pointwise almost everywhere, and by integrating by parts in (3.1) we deduce that Here f (z + ) := lim We now describe the solutions to (3.3), equivalently (3.1).
3.1. Representation via a fundamental system. Due to the existence and uniqueness theorem for linear first-order systems with locally integrable coefficients (see, e.g., [21]), for all \lambda \in \BbbR the first-order system has a fundamental solution system so that any solution to - (a 0 u \prime ) \prime = \lambda \rho 0 u in Y 0 is a linear combination of v 1 (\lambda , \cdot ) and v 2 (\lambda , \cdot ), and cf. Remark 2.3. Note that the associated Wronskian of the system is constant: for some c 1 , c 2 \in \BbbC . Substituting the representation (3.7) into the second and third equations of (3.3) leads to the system For the existence of a nontrivial solution (c 1 , c 2 ) to (3.8), and therefore nontrivial u in (3.3), the value \lambda must necessarily solve the equation From the relation (3.9) we can deduce much more about the limit spectrum. Setting \lambda k (\theta ), k \in \BbbN , \theta \in [0, 2\pi ), to the kth eigenvalue of A \theta labeled according to the min-max principle, we define E k : [0, 2\pi ) \rightar [0, \infty ) to be the kth spectral band function given by \theta \mapsto \rightar \lambda k (\theta ). The name``spectral band function"" comes from the (clear) characterization: \bigcup We shall prove below the following result about the nature of the spectral band functions.
Theorem 3.1. 1. The functions E k , k \in \BbbN , are continuous and even around \theta = \pi : The functions E 2m - 1 (\cdot ), m \in \BbbN , are strictly increasing on (0, \pi ). 3. The functions E 2m (\cdot ), m \in \BbbN , are strictly decreasing on (0, \pi ). 4. The spectral bands are given by the following intervals: Let us focus on claim 4 of the above theorem. It informs us that the interval \bigl( ) is a spectral gap if and only if \lambda = \lambda 2m - 1 (\pi ) (respectively, \lambda = \lambda 2m (0)) is a simple eigenvalue of the antiperiodic (respectively, periodic) limit problem (3.2). 2 We now characterize when the eigenvalues of the periodic, antiperiodic problems are degenerate, i.e., have multiplicity two, in terms of the fundamental system (v 1 , v 2 ). At such points \lambda the spectral bands touch and there are no gaps.
Proof of Proposition 3.2. Let us consider \lambda n (0), the case \lambda n (\pi ) is similar. Necessity follows by noting that if \lambda = \lambda n (0) has multiplicity 2 then v i (\lambda , \cdot ), i = 1, 2, are linear combinations of the orthogonal periodic eigenfunctions to the limit problem (3.11). In particular the conditions (3.11) follows from the initial values v 2 (\lambda , 0) = 0 and (a 0 v \prime 1 )(\lambda , 0) = 0. For sufficiency, by (3.13) both the fundamental solutions satisfy the limit spectral problem; i.e., they are linearly independent eigenfunctions of \lambda = \lambda n (0). The proof of Theorem 3.1 readily follows, by arguing, for example, as in [9, Chapter 2.3], from the following further analysis on the function D given by (3.9).
Lemma 3.4. The function D is analytic. Furthermore, the following assertions hold.
Next, we suppose that D \prime (\lambda n (0)) = 0 and prove that (3.11) holds. As \lambda n (0) is a root of D(\cdot ) 2 -4, we deduce from (3.20) that Therefore v 2 2 = v 2 (\lambda n (0), h) 2 = 0, and Since Proof of (c). We shall prove that Then, as v 1 (\lambda n (0), \cdot ) and v 2 (\lambda n (0), \cdot ) are linearly independent it follows from (3.21) and the H\" older inequality that Let us prove (3.21). First we make some preliminary calculations. By (3.16) and (3.13), which holds since D \prime (\lambda n (0)) = 0 (cf. (b)) and Remark 3.3, we compute (dropping the argument (\lambda n (0), h) as above) We now are going to differentiate both sides of (3.17) with respect to \lambda . Half of the terms will immediately been seen to be zero. Indeed, if we take the first term in the right-hand side of (3.17), differentiate it, and evaluate it at (\lambda n (0), h), then because \alpha = 0 (cf. (3.19) and (3.13)) we deduce that The same is true for all the other terms. Therefore, differentiating (3.17) and bearing in mind (3.22), we compute (after a little bit more algebra) that From (3.22) we see that \partial \lambda v 2 \geq 0 and so it follows from (3.23) that We complete the proof of (3.21) (cf. (3.22)) if we can prove that Multiplying (3.18) by v 1 (\lambda , s) and utilizing (3.6) gives Then, differentiating both sides of the above equation with respect to \lambda and evaluating at (\lambda n (0), h) gives Upon utilizing (3.13), we deduce that (3.24) holds, and the proof of (c) follows. The proofs of (d) and (e) are similar to that of (b) and (c).

Representation via a spectral decomposition. Consider the operator \
A \theta defined on L 2 \rho 0 (Y 0 ) and associated to the form in the sense of procedure described in section 2. By virtue of the fact that the operator \Ã \theta has compact resolvent, its L 2 \rho 0 (Y 0 )-orthonormal sequence of eigenfunctions \{ \Phi (n) \theta \} n\in \BbbN is complete in the space L 2 \rho 0 (Y 0 ). We denote by \mu n (\theta ), n \in \BbbN , the eigenvalues of \Phi

Multiplying the first equation in (3.3) by \Phi
(n) \theta and integrating by parts we have Therefore, upon performing a spectral decomposition of u in terms of \Phi we see that In particular, one has u(0) = \sum n\in \BbbN \zeta n \Phi  The converse statement is also true. Indeed, suppose that for some \theta \in [0, 2\pi ) the value \lambda satisfies (3.26), then we find that Since, by assumption, it follows that \sum n | \zeta n | 2 < \infty , \sum n \mu n (\theta )| \zeta n | 2 < \infty , that is, the function u = \sum n\in \BbbN \zeta n \Phi (n) \theta belongs to H 1 \theta (Y 0 ) and, consequently, to V \theta when extended by the constant u(h) into Y 1 . Moreover, direct calculation shows that \lambda and u satisfy (3.1). Hence, we have shown that \lambda \in \bigcup \theta \sigma (A \theta ).
4. Asymptotics of the defect eigenvalue problem. Suppose \lambda \varepsi , u \varepsi is an eigenvalue-eigenfunction pair for the defect problem, that is,  In this section we study the behavior with respect to \varepsi of the eigenvalues \lambda \varepsi and eigenfunctions u \varepsi , using asymptotic expansions. We show that, up to the leading order, the values of \lambda \varepsi are described by an eigenvalue of the weighted Neumann--Laplacian on the defect D; see (4.6) below. More precisely, we show that for each eigenvalue \lambda 0 of (4.6) in a gap of \bigcup \theta \sigma (A \theta ), there exists a sequence of eigenvalues \lambda \varepsi of (4.1) converging to \lambda 0 . However, it remains unclear whether every accumulation point of \lambda \varepsi inside a gap of \bigcup \theta \sigma (A \theta ) belongs to the spectrum of (4.6). We seek asymptotic expansions for the eigenvalues \lambda \varepsi and eigenfunctions u \varepsi of (4. \left\{ for all (4.9) z \in \scrI \varepsi := \bigl\{ z \in \BbbZ : z \geq \lceil d + \rceil \varepsi or z \leq \lfloor d -\rfloor \varepsi -1 \bigr\} (that is, z \in \scrI \varepsi if and only if the intersection of \varepsi (Y + z) and D is empty). The assertion (4.7) implies that a 1 w \prime 0 \equiv 0 on Y 1 + z and therefore w 0 is constant on each such interval. By the second equation of (4.8) and the fact w 0 is constant on each interval Y 1 + z, the function a 1 w \prime 2 has the form Combining (4.10), the fact that w 0 is constant on Y 1 +z and the first and last equations of (4.8) implies that for all z \in \scrI \varepsi , one has (4.11) The problem (4.11) fully governs the behavior of w 0 in \BbbR \setminu (\lfloor d -\rfloor \varepsi -1, \lceil d + \rceil \varepsi ). We can utilize the fundamental system (v 1 , v 2 ) from section 3.1 to quantitatively characterize w 0 . Indeed, since in each cell Y + z any solution to the first equation in (4.11) is a linear combination of v 1 and v 2 , one has for constants l z , m z , z \in \scrI \varepsi , where the expression on Y 1 + z follows from the second condition in (4.11). Using (3.5), the continuity of w 0 and the jump of the coderivative condition from (4.11), it is not difficult to derive the following recurrence relation: (4.13) Now, recalling the Wronskian property (3.6), we find that the characteristic polynomial q of the matrix in (4.13) is (cf. (2.10)) The roots \mu 1 , \mu 2 of q satisfy the identity \mu 1 \mu 2 = 1 and the nature of w 0 as it varies from one period to the next is determined by the quantity v 1 then the roots \mu 1 , \mu 2 are complex conjugate with | \mu 1 | = | \mu 2 | = 1 and solutions w 0 are described by the linear span of two quasi-periodic functions with phase difference \pi .

2.
For sufficiently small \varepsi there exist (simple) eigenvalues \lambda \varepsi of A \varepsi D such that | \lambda \varepsi -\lambda 0 | \leq C 1 \varepsi 1/2 . 3. For sufficiently small \varepsi the function u \varepsi ,ap is an approximate eigenfunction of A \varepsi D , in the sense that there exists an \varepsi -independent constant C 2 > 0 and c \varepsi j \in \BbbR such that where the set J \varepsi is defined by (2.8), and u \varepsi ,j are appropriate eigenfunctions of A \varepsi D . Proof. Claim 1 of the theorem follows from (4.28) and the fact that due to (4.18), (4.25), and (4.29). Claim 2 follows by noting that the essential spectra of A \varepsi D and A \varepsi coincide and that \sigma (A \varepsi ) = \sigma ess (A \varepsi ) converges, as \varepsi \rightar 0, to \bigcup \theta \sigma (A \theta ), which \lambda 0 does not belong to. To prove claim 3, one can argue as in [19], or [11, section 11.1], using (4.28) and a spectral decomposition of u \varepsi ,ap with respect to the operator A \varepsi D . 4.4. Improvement of the error bound. It is clear from the construction of u \varepsi ,ap that the main error term of order \varepsi 1/2 comes from what is conventionally called boundary layer, near the endpoints of the defect D. In fact, one can improve the error bound (4.28) by``attaching"" the \varepsi -periodic structure to the defect in an appropriate way, thereby preventing the appearance of the boundary effect. Our approach is based on the behavior of the function w 0 ; see the observation made in the beginning of section 4.2 preceding the adjustment of w 0 . We provide the detailed construction only at the right end of the defect D. The construction at the left end is completely analogous.
(We remind the reader that the notation a \varepsi D , \rho \varepsi D , \Omega \varepsi 0 and \Omega \varepsi 1 has to be redefined accordingly to the above construction in each case).
In the case if w 0 has no extrema inside the soft component at least on one of the intervals ( - \infty , d -/\varepsi ] or [d + /\varepsi , \infty ), the term u 1 is nonzero. Then, similarly to (4.28), we obtain an improved estimate The improved estimates for the error term immediately imply the following statement.
Remark 4.3. In the above theorem the attached structures do not need to be periodic extensions of each other. In case of``nonmatching"" periodic structures on each side of the defect the essential spectrum of the resulting operator is exactly the same as in the purely periodic case without the defect. This can easily be seen by considering Weyl's sequences in each of the cases.

Extreme localization of defect eigenfunctions.
The method of asymptotic expansions allows us to show that for any eigenvalue \lambda 0 of A N,D (cf. (2.6), (4.6)) in a gap of \bigcup \theta \sigma (A \theta ) there exists a sequence of eigenvalues of A \varepsi D converging to \lambda 0 . In this section we provide a statement on the rate of decay of eigenfunctions of A \varepsi D outside the defect. Namely, the fact that one-dimensional problems admit an explicit form of solutions in terms of the fundamental system allows us to show that the eigenfunctions u \varepsi decay at an accelerated exponential rate outside of the defect, which is claim 2 of Theorem 2.4.
Recalling, from section 3.1, the fundamental solutions v 1 , v 2 of (cf. (3.5)) we shall prove in the second half of this section the following property.
6. Resolvent estimates for the problem without defect. In this section we study the behavior of the unperturbed periodic operator A \varepsi in the operator norm as \varepsi \rightar 0. In particular, we construct a full asymptotic expansions for the resolvent of A \varepsi using a version of the asymptotic framework developed in [7]; see Theorem 6.2 below. This directly implies the order-sharp operator norm resolvent convergence estimate, uniform in \theta , formulated in Theorem 2.2. The latter, in turn, implies the uniform in \theta convergence, as \varepsi \rightar 0, of the spectral band functions \lambda \varepsi n (\theta ) to \lambda n (\theta ), n \in \BbbN , which is also order-sharp.
In particular, this indirectly implies, since \lambda n is the uniform limit of continuous functions, that \lambda n is continuous in \theta . A direct proof of this fact can be arrived at by the definition of the operators A \theta and the continuity properties (in the Hausdorff sense) of their domains D(A \theta ); see [8,Appendix B].