Extreme localisation of eigenfunctions to one-dimensional high-contrast periodic problems with a defect
aa r X i v : . [ m a t h . SP ] F e b Extreme localisation of eigenfunctions toone-dimensional high-contrast periodicproblems with a defect
Mikhail Cherdantsev , Kirill Cherednichenko , and Shane Cooper Cardiff School of Mathematics, Senghennydd Road, Cardiff, CF24 4AG Department of Mathematical Sciences, University of Bath, Claverton Down, Bath,BA2 7AY, UKJuly 21, 2018
Abstract
Following a number of recent studies of resolvent and spectral convergence of non-uniformlyelliptic families of differential operators describing the behaviour of periodic composite mediawith 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 spectrumof the purely periodic case without the defect and its limit, as the period ε goes to zero, has aband-gap structure. We consider a sequence of eigenvalues λ ε that are induced by the defectand converge to a point λ located in a gap of the limit spectrum for the periodic case. Weshow that the corresponding eigenfunctions are “extremely” localised to the defect, in the sensethat the localisation exponent behaves as exp( − ν/ε ) , ν > , which has not been observed inthe existing literature. As a consequence, we argue that λ is an eigenvalue of a certain limitoperator 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 ofpropagating waves that are localised to the defect. We also show that the unperturbed operatorsare norm-resolvent close to a degenerate operator on the real axis, which is described explicitly. Keywords
High-contrast homogenisation · Wave localisation · Spectrum · Decay estimates
The question of whether a macroscopic perturbation of material properties in a periodic mediumor structure (periodic composite) induces the existence of a localised solution (bound state) tothe 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 solutioncan have either an advantageous or undesirable effect on the behaviour of systems containing therelated composite medium as a component. For example, in the context of photonic (phononic)crystal fibres, 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 [13, 17]. Inthe mathematics literature, proofs of the existence or non-existence of such a localised solution havebeen carried out using the tools of the classical asymptotic analysis of the governing equations andspectral analysis of operators generated by the governing equations in various “natural” functionspaces. The choice of the concrete class of equations and functions under study is usually motivatedby the applications in mind, and several works that have marked the development of the relatedanalytical techniques cover a wide range of operators and their relatively compact perturbations, e.g. [21], [3], [2], [9]. 1he present work is a study of localisation properties for a class of composite media that hasbeen the subject of increasing interest in the mathematics and physics literature recently, in view ofit relation to the so-called metamaterials, e.g. manufactured composites possessing negative refrac-tion 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 aninfinitely increasing number of gaps (lacunae) opening in the spectrum, in the limit of the smallratio ε between the period and the overall size of the composite. This analytical feature, analogousto the spectral property of multi-dimensional high-contrast periodic composites shown in [19], pro-vides a mathematical recipe for the use of such materials in physics context or technologies wherethe presence of localised modes (generated by defects in the medium) has important practical im-plications. In the physical context of photonic crystal fibres and within the mathematical setupof multi-dimensional 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 differentialoperator with a finite-size perturbation (defect) was derived. It was shown that for eigenvalues λ ingaps of the spectrum of the (two-scale) operator representing the leading-order term of this asymp-totics, there are sequences of eigenvalues of the finite-period problems that converge to λ as ε → . The subsequent works [5], [6] developed a multiscale version of Agmon’s approach [1] and provedthat the corresponding eigenfunctions of the limit operator decay exponentially fast away from thedefect. An important technical assumption in all these works is that the low-modulus inclusions inthe composite have a positive distance to the boundary of the period cell, which is not possible tosatisfy in one dimension.In the more recent paper [8], a family of non-uniformly elliptic periodic one-dimensional problemswith high contrast was studied, which in practically relevant situations corresponds to a stratifiedcomposite with alternating layers of homogeneous media with highly contrasting material properties.It was shown that the spectra of the corresponding operators converge, as ε →
0, to the band-gapspectrum 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 − ( a εD u ′ ) ′ , a εD > , where a εd takes values of order one on D and ε -periodic ( ε >
0) in R \ D with alternating values oforder one and ε . As was mentioned in [8, Section 5.1], a formal analysis suggests that the rate ofdecay of eigenfunctions localised 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 valueas ε → . The goal of the present work is to provide a rigorous proof of this property, formulatedbelow as Theorem 2.4. In view of the above discussion, this new localisation property can be seen asa consequence of two features of the underlying periodic composite: loss of uniform ellipticity (viathe presence of soft inclusions in the moderately stiff matrix material) and the geometric conditionof the matrix material components being separated by the inclusions.In addition to our main result, we formulate (Section 3) a new characterisation of the limitspectrum for the unperturbed family of problems in the whole space discussed in [8] and strengthen(Section 7) the result of [8] by proving an order-sharp norm-resolvent convergence estimate forthis family (Theorem 2.2). In particular, this new estimate implies order-sharp uniform convergenceestimates, as ε → , for the related family of parabolic problems, via the norm-resolvent convergenceof the corresponding operator semigroups. For ε, h ∈ (0 , , we introduce the setsΩ ε := [ z ∈ Z ( εz, εz + εh ) , and Ω ε := [ z ∈ Z ( εz + εh, εz + ε ) = R \ Ω ε , Y := (0 , h ), Y := ( h, Y := (0 , ε -periodic functions a ε ( x ) := ( ε a ( xε ) , x ∈ Ω ε ,a ( xε ) , x ∈ Ω ε , ρ ε ( x ) = ρ ( xε ) , ρ ( y ) := ( ρ ( y ) , y ∈ Y ,ρ ( y ) , y ∈ Y , (2.1)for a j , a − j , ρ j , ρ − j ∈ L ∞ ( Y j ), j = 1 ,
2, periodic with period 1. It is convenient to set a ≡ Y and a ≡ Y , thus we can write, for example, a ε ( x ) = ε a ( x/ε ) + a ( x/ε ).For a positive Lebesgue-measurable function w on a Borel set B ⊂ R , such that w, w − ∈ L ∞ ( B ),we employ the notation L w ( B ) to indicate that the space L ( B ) is equipped with the inner product( · , · ) w := Z B w | · | . For a bilinear form β : H ( R ) × H ( R ) → R , the (self-adjoint) operator A associated to β is denselydefined in L w ( R ) by the action Au = f, where for a given u ∈ H ( R ), the function f ∈ L ( R ) is thesolution to the integral identity β ( u, v ) = Z R wf v ∀ v ∈ H ( R ) . For the bilinear form β ε ( u, v ) := Z R a ε u ′ v ′ , u, v ∈ H ( R ) , we consider A ε , the operator defined in L ρ ε ( R ) and associated to β ε . The spectrum σ ( A ε ) of A ε isabsolutely continuous and, by introducing the rescaled Floquet-Bloch transform U ε , see (7.69), wenote that σ ( A ε ) admits the representation σ ( A ε ) = [ θ ∈ [0 , π ) σ ( A εθ ) , where σ ( A εθ ) is the spectrum of the L ρ ( Y ) densely-defined self-adjoint operator A εθ associated to theform β εθ ( u, v ) := Z Y (cid:0) a + ε − a (cid:1) u ′ v ′ , acting in the space H θ ( Y ) of functions u ∈ H ( Y ) that are θ -quasiperiodic, i.e. such that u ( y ) =exp(i θy ) v ( y ) , y ∈ Y, for some 1-periodic function v ∈ H ( Y ). For each ε, θ, the operator A εθ hascompact resolvent and consequently its spectrum σ ( A εθ ) is discrete.Consider the space V θ := (cid:8) u ∈ H θ ( Y ) : u ′ = 0 on Y (cid:9) and its closure in L ρ ( Y ) , which we denote by V θ . We introduce the densely defined operators A θ : V θ → L ρ ( Y ) given by A θ u = f for u, f such that Z Y a u ′ v ′ = Z Y ρf v ∀ v ∈ V θ . (2.2)For each θ, the operator A θ has compact resolvent and so σ ( A θ ) is discrete. In a recent work [8],see Section 7, the spectrum σ ( A ε ) was shown to converge in the Hausdorff sense to the union of thespectra of the operators A θ , i.e. lim ε → σ ( A ε ) = [ θ ∈ [0 , π ) σ ( A θ ) . (2.3) Remark 2.1. S θ ∈ [0 , π ) σ ( A θ ) can be seen as the spectrum of a certain operator A unitary equivalentto the direct integral of operators R θ A θ , see Appendix A for the details.
3n Section 7, we construct infinite-order asymptotics (as ε →
0) for the resolvents of A εθ , uniformin θ, with respect to the H norm and, in particular, prove the following refinement of the resultestablished in [8]: Theorem 2.2.
The operator A εθ norm-resolvent converges to A θ , uniformly in θ , at the rate ε .More precisely, there exists a constant C > such that (cid:13)(cid:13) ( A εθ + 1) − f − ( A θ + 1) − f (cid:13)(cid:13) L ρ ( Y ) ≤ Cε || f || L ρ ( Y ) ∀ θ ∈ [0 , π ) . Consequently, since the spectra σ ( A εθ ) and σ ( A θ ) are discrete, we have the following result: foreach n ∈ N there exists a constant c n > (cid:12)(cid:12) λ εn ( θ ) − λ n ( θ ) (cid:12)(cid:12) ≤ c n ε ∀ θ ∈ [0 , π ) , ε ∈ (0 , . Here, { λ εn ( θ ) } n ∈ N , { λ n ( θ ) } n ∈ N are the eigenvalue sequences of A εθ , A θ , respectively, labelled in theincreasing order . From this theorem it follows that for sufficiently small ε , the spectrum σ ( A ε ) hasgaps if the set S θ σ ( A θ ) contains gaps. In Section 3 we demonstrate that this set contains infinitelymany gaps. Furthermore, we demonstrate that λ ∈ S θ σ ( A θ ) if and only if the inequality (cid:12)(cid:12)(cid:12) v ( h ) + ( a v ′ )( h ) − λv ( h ) Z Y ρ (cid:12)(cid:12)(cid:12) ≤ v and v are the ( λ -dependent) solutions of − ( a v ′ j ) ′ = λρ v j on Y , j = 1 , , subject to the conditions v (0) v (0)( a v ′ )(0) ( a v ′ )(0) ! = ! . Remark 2.3.
Note that any solution u of − ( a u ′ ) ′ = λρ u is absolutely continuous and so is itsco-derivative a u ′ . Hence, their value at any point y is well defined (unlike the value of a or u ′ ingeneral). This explains the use of notation ( a v ′ j )( y ) , which we will hold to throughout the paper. Next, we introduce d − , d + ∈ R and on the set D = ( d − , d + ) replace the coefficients (2.1) by someuniformly positive and bounded functions a D , ρ D , namely we consider a εD ( x ) := a D ( x ) , x ∈ D,a ( xε ) , x ∈ Ω ε \ D,ε a ( xε ) , x ∈ Ω ε \ D, ρ εD ( x ) := ρ D ( x ) , x ∈ D,ρ ( xε ) , x ∈ Ω ε \ D,ρ ( xε ) , x ∈ Ω ε \ D. We shall study the spectrum of the operator A εD defined in L ρ εD ( R ) and associated to the form β εD ( u, v ) := Z R a εD u ′ v ′ , u, v ∈ H ( R ) . (2.4)As this operator arises from a compact perturbation of the coefficients of A ε , it is well-known, see e.g. [9], that the essential spectra of A εD and A ε coincide. For eigenvalues situated, for small valuesof ε, in the gaps of the essential spectrum of A εD (equivalently, in the gaps of the essential spectrumof σ ( A ε )), we expect the eigenfunctions to be localised around the defect, and therefore we areinterested in the analysis of eigenfunctions of A εD corresponding to eigenvalues that are located inthe gaps of the limit spectrum S θ σ ( A θ ). We show that for the sequence of point spectra σ p ( A εD ) ofthe operators A εD , the set of accumulation points as ε → Notice that all the eigenvalues are simple due to the 1-dimensional nature of the corresponding problem. ε → σ ( A ε ) is given by the intersection of the set R \ S θ σ ( A θ ) with the spectrum of theoperator A N ,D defined in L ρ D ( D ) and associated to the form β N ,D ( u, v ) := Z D a D u ′ v ′ , u, v ∈ H ( D ) . (2.5)The functions from the domain of A N ,D satisfy the Neumann condition on the boundary of D. Conversely, if we choose the defect D so that the spectrum σ ( A N ,D ) has a non-empty intersectionwith R \ S θ σ ( A θ ), then for sufficiently small ε the operator A εD has non-empty point spectrum. No-tice that we can always choose a D , ρ D , d − and d + such that this is true. Moreover, we demonstratethat for eigenvalue sequences that converge to a point in R \ S θ σ ( A θ ) the corresponding eigenfunc-tions are localised to a small neighbourhood of the defect. Namely, the eigenfunctions u ε exhibit ac-celerated exponential decay outside the defect in the sense that the function exp (cid:0) dist( x, D ) ν/ε (cid:1) u ε ( x ) ,x ∈ R , is an element of L ( R \ D ) for sufficiently small ε, where the value ν is determined by thedistance of the limit point of λ ε to the set S θ σ ( A θ ). These results are collated in the followingtheorem, which we prove in Sections 4, 5, 6. Theorem 2.4.
1. For every λ ∈ σ ( A N ,D ) \ (cid:0)S θ σ ( A θ ) (cid:1) (which is always simple) there exist a unique sequenceof simple eigenvalues λ ε of A εD converging to λ and constants C , C > such that | λ ε − λ | ≤ C ε, k u ε − u k L ( D ) ≤ C ε / , k u ε k L ( R \ D ) ≤ C ε / , (2.6) where u and u ε are the eigenfunctions of λ and λ ε respectively.2. Conversely, for any sequence λ ε ∈ σ p ( A εD ) such that λ ε → λ / ∈ lim ε → σ ( A ε ) = S θ σ ( A θ ) , onehas λ ∈ σ ( A N ,D ) .3. Furthermore, the L ( R ) -normalised eigenfunctions u ε of A εD corresponding to the eigenvalues λ ε are localised to the defect in the following sense: for ν > , let g ν/ε denote the exponentiallygrowing function g ν/ε ( x ) := ( , x ∈ D, exp (cid:0) νε dist( x, D ) (cid:1) , x ∈ R \ D, (2.7) and take µ to be the smallest by the absolute value root of the quadratic function q ( µ ) := µ − (cid:18) v ( h ) + ( a v ′ )( h ) − λ v ( h ) Z Y ρ (cid:19) µ + 1 . (2.8) Then, for sufficiently small values of ε the function g ν/ε u ε is an element of L ( R ) for all ν < | ln | µ || . Here we quantitatively characterise the spectrum ( cf. (2.3)) [ θ ∈ [0 , π ) σ ( A θ ) By this we mean asymptotically unique, i.e. if there are two sequences λ ε and λ ′ ε converging to λ then necessary λ ε = λ ′ ε for small enough ε . λ ∈ [0 , ∞ ) and u ∈ V θ = (cid:8) v ∈ H θ ( Y ) : v ′ ≡ Y (cid:9) such that h Z a u ′ v ′ = λ Z ρuv ∀ v ∈ V θ . (3.9)By taking test functions v ∈ C ∞ ( Y ) we deduce that u | Y is a weak solution to the equation − ( a u ′ ) ′ = λρ u (3.10)on Y . For L ∞ -functions a and ρ the equation (3.10) holds pointwise almost everywhere and byintegrating by parts in (3.9) we deduce that( a u ′ )( h − ) v ( h ) − ( a u ′ )(0 + ) v (0) = λ Z Y ρ uv ∀ v ∈ V θ . Here f ( z + ) := lim x ց z f ( x ) , and f ( z − ) := lim x ր z f ( x ) for a function f, whenever the corresponding limitexists. Since any element v ∈ V θ satisfies v ( y ) = e i θ v (0) , y ∈ Y , the above observations imply that u satisfies (3.9) if, and only if, w = u | Y ∈ H ( Y ) is a weak solution of the problem − ( a u ′ ) ′ = λρ u in Y ,u ( h ) = e i θ u (0) ,e − i θ ( a u ′ )( h − ) − ( a u ′ )(0 + ) = λu (0) Z Y ρ . (3.11)We now describe the solutions to (3.11), equivalently (3.9). Due to the existence and uniqueness theorem for linear first order systems with locally integrablecoefficients, see e.g. [18], there exist a fundamental system of solutions v and v to the equation − ( a u ′ ) ′ = λρ u such that v (0) v (0)( a v ′ )(0) ( a v ′ )(0) ! = ! , (3.12)cf. Remark 2.3. The Wronskian of the system is constant: v ( y )( a v ′ )( y ) − v ( y )( a v ′ )( y ) = 1 , y ∈ Y , (3.13)and any solution u to (3.11) is of the form u = c v + c v for some c , c ∈ C . The substitution of the above representation for u in terms of v , v into the second and thirdequations of (3.11) leads to the system v ( h ) − e i θ v ( h )( a v ′ )( h ) − e i θ λ R Y ρ ( a v ′ )( h ) − e i θ ! (cid:18) c c (cid:19) = (cid:18) (cid:19) . (3.14)For the existence of a non-trivial solution ( c , c ) to (3.14), and therefore non-trivial u in (3.11), thevalue λ must necessarily solve the equation2 cos( θ ) = v ( h ) + ( a v ′ )( h ) − λv ( h ) Z Y ρ . Hence, the set ( cf. (2.3)) [ θ ∈ [0 , π ) σ ( A θ )6onsists of all non-negative λ such that the following inequality holds: (cid:12)(cid:12)(cid:12)(cid:12) v ( h ) + ( a v ′ )( h ) − λv ( h ) Z Y ρ (cid:12)(cid:12)(cid:12)(cid:12) ≤ . (3.15)As noted above, the functions v , v depend on the spectral parameter λ . We demonstrate theimplications of this dependence for the structure of the limit spectrum through the following simpleexample. Assume that a , ρ and ρ are equal to unity on their support, then v = cos( √ λy ), v = √ λ sin( √ λy ), and the limit spectrum is given by (cid:12)(cid:12)(cid:12)(cid:12) √ λh ) − √ λ sin( √ λh )(1 − h ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ . In particular we see that the bands of the limit spectrum become very narrow as λ → ∞ . Consider the operator ˜ A θ defined on L ρ ( Y ) and associated to the form˜ β θ ( u, v ) := Z Y a u ′ v ′ , u, v ∈ H θ ( Y ) , in the sense of procedure described in Section 2. By virtue of the fact that the operator ˜ A θ hascompact resolvent, its L ρ ( Y )-orthonormal sequence of eigenfunctions { Φ ( n ) θ } n ∈ N is complete in thespace L ρ ( Y ). We denote by µ n ( θ ), n ∈ N , the eigenvalues of Φ ( n ) θ ∈ H θ ( Y ): Z Y a (cid:0) Φ ( n ) θ (cid:1) ′ v ′ = µ n ( θ ) Z Y ρ Φ ( n ) θ v ∀ v ∈ H θ ( Y ) . (3.16)Multiplying the first equation in (3.11) by Φ ( n ) θ and integrating by parts we have λ Z Y ρ u Φ ( n ) θ = − Z Y ( a u ′ ) ′ Φ ( n ) θ = − (cid:0) ( a u ′ )( h − )Φ ( n ) θ ( h ) − ( a u ′ )(0 + )Φ ( n ) θ (0) (cid:1) + Z Y a u ′ (cid:0) Φ ( n ) θ (cid:1) ′ = − (cid:0) e − i θ ( a u ′ )( h − ) − ( a u ′ )(0 + ) (cid:1) Φ ( n ) θ (0) + µ n ( θ ) Z Y ρ u Φ ( n ) θ . The third equation in (3.11) implies (cid:0) µ n ( θ ) − λ (cid:1) Z Y ρ u Φ ( n ) θ = λu (0)Φ ( n ) θ (0) Z Y ρ Therefore, upon performing a spectral decomposition of u in terms of Φ ( n ) θ , i.e. setting u = X n ∈ N ζ n Φ ( n ) θ , ζ n = Z Y ρ u Φ ( n ) θ , we see that ζ n = λµ n ( θ ) − λ u (0)Φ ( n ) θ (0) Z Y ρ , n ∈ N . In particular, one has u (0) = P n ∈ N ζ n Φ ( n ) θ (0). Thus we arrive at the following alternative descriptionof the limit spectrum: λ ∈ S θ σ ( A θ ) if and only if there exist θ ∈ [0 , π ) such that X n ∈ N λµ n ( θ ) − λ (cid:12)(cid:12) Φ ( n ) θ (0) (cid:12)(cid:12) = (cid:18)Z Y ρ (cid:19) − . Asymptotics of the defect eigenvalue problem
Suppose λ ε , u ε is an eigenvalue-eigenfunction pair for the defect problem, that is − ( a εD u ′ ε ) ′ = λ ε ρ εD u ε on R , (4.17)where u ε is continuous, subject to the interface conditions a D u ′ ε (cid:12)(cid:12) D = a εD u ′ ε (cid:12)(cid:12) R \ D on { d − , d + } . (4.18)and a u ′ ε (cid:12)(cid:12) Ω ε \ D = ε a u ′ ε (cid:12)(cid:12) Ω ε \ D on (cid:8) x ∈ R \ D : x = ε ( z + h ) or x = εz for some z ∈ Z (cid:9) . (4.19)In this section we study the behaviour with respect to ε of the eigenvalues λ ε and eigenfunctions u ε , using asymptotic expansions. We show that, up to the leading order, the values of λ ε aredescribed by an eigenvalue of the weighted Neumann-Laplacian on the defect D , see (4.22) below.More precisely, we show that for each eigenvalue λ of (4.22) in a gap of S θ σ ( A θ ), there exists asequence of eigenvalues λ ε of (4.17) converging to λ . However, it remains unclear whether everyaccumulation point of λ ε inside a gap of S θ σ ( A θ ) belongs to the spectrum of (4.22). We addressthis question in Section 5, where we argue that the eigenmodes u ε are asymptotically localised tothe defect. The latter observation implies compactness of the sequences of eigenmodes u ε , thusestablishing asymptotic one to one correspondence between the eigenvalues of (4.17) and (4.22) inthe gaps of S θ σ ( A θ ).We seek asymptotic expansions for the eigenvalues λ ε and eigenfunctions u ε of (4.17)–(4.19) inthe form λ ε = λ + ελ + ε λ . . . , (4.20)with u ε ( x ) = ( u ( x ) + εu ( x ) + ε u ( x ) + . . . , x ∈ ( d − , d + ) ,w ( xε ) + ε w ( xε ) + . . . , x ∈ ( −∞ , d − ) ∪ ( d + , ∞ ) . (4.21)We assume that functions w i , u i , i = 0 , , , . . . , are continuous. Substituting (4.20), (4.21) into(4.17) and (4.18) and equating the ε -coefficient on the defect gives ( − ( a D u ′ ) ′ = λ ρ D u on ( d − , d + ) ,a D u ′ | D = 0 on { d − , d + } , (4.22)that is, λ is an eigenvalue of the weighted Neumann-Laplace operator A N ,D on the defect, cf. (2.5). Note that this is true regardless of whether d − , d + belong to Ω ε or Ω ε . We fix u by setting k u k L ρD ( D ) = 1.For c ∈ R , let ⌊ c ⌋ ε and ⌈ c ⌉ ε denote the largest integer z such that εz ≤ c and the smallest integer z such that c ≤ εz, respectively. Substituting (4.20), (4.21) into (4.17), (4.19) and comparing thecoefficients for different powers of ε in the resulting expression yields ( − ( a w ′ ) ′ = 0 , on Y + z,a w ′ (cid:12)(cid:12) Y + z = 0 , on { z + h, z + 1 } , (4.23)and − ( a w ′ ) ′ = λ ρ w , on Y + z, − ( a w ′ ) ′ = λ ρ w , on Y + z,a w ′ (cid:12)(cid:12) Y + z = a w ′ (cid:12)(cid:12) Y + z , on { z + h, z + 1 } , (4.24)for all z ∈ I ε := (cid:8) z ∈ Z : z ≥ ⌈ d + ⌉ ε or z ≤ ⌊ d − ⌋ ε − (cid:9) . (4.25)8he assertion (4.23) implies that a w ′ ≡ Y + z and therefore w is constant on each suchinterval. By the second equation of (4.24), and the fact w is constant on each interval Y + z , thefunction a w ′ has the form( a w ′ )( y ) = ( a w ′ )( z + h ) − λ w ( z + h ) y Z z + h ρ , y ∈ Y + z. (4.26)Combining (4.26), the fact w is constant on Y + z and the first and last equations of (4.24) impliesthat for all z ∈ I ε one has − ( a w ′ ) ′ = λ ρ w , on Y + z,w ≡ w ( z + h ) = w ( z + 1) , on Y + z, ( a w ′ ) (cid:0) ( z + 1) + (cid:1) − ( a w ′ ) (cid:0) ( z + h ) − (cid:1) = − λ w ( z + h ) R Y ρ . (4.27)The problem (4.27) fully governs the behaviour of w in R \ ( ⌊ d − ⌋ ε − , ⌈ d + ⌉ ε ). We can utilise thefundamental system ( v , v ) from Section 3.1 to quantitatively characterise w . Indeed, since in eachcell Y + z any solution to the first equation in (4.27) is a linear combination of v and v , one has w ( y ) = ( l z v ( y − z ) + m z v ( y − z ) , y ∈ Y + z,l z v ( h ) + m z v ( h ) , y ∈ Y + z, (4.28)for constants l z , m z , z ∈ I ε , where the expression on Y + z follows from the second condition in(4.26). Using the continuity of w and the jump of the co-derivative condition from (4.27) it is notdifficult to derive the following recurrence relation: (cid:18) l z +1 m z +1 (cid:19) = v ( h ) v ( h )( a v ′ )( h ) − λ v ( h ) R Y ρ ( a v ′ )( h ) − λ v ( h ) R Y ρ ! (cid:18) l z m z (cid:19) . (4.29)Now, recalling the Wronskian property (3.13), we find that the characteristic polynomial q of thematrix in (4.29) is ( cf. (2.8)) q ( µ ) = µ − (cid:16) v ( h ) + ( a v ′ )( h ) − λ v ( h ) Z Y ρ (cid:17) µ + 1 . The roots µ , µ of q satisfy the identity µ µ = 1 and the nature of w as it varies from one periodto the next is determined by the quantity v ( h ) + ( a v ′ )( h ) − λ v ( h ) R Y ρ . Namely, if ( cf. (3.15)) (cid:12)(cid:12)(cid:12)(cid:12) v ( h ) + ( a v ′ )( h ) − λ v ( h ) Z Y ρ (cid:12)(cid:12)(cid:12)(cid:12) ≤ , then the roots µ , µ are complex conjugate with | µ | = | µ | = 1 and solutions w are described bythe linear span of two quasi-periodic functions with phase difference π . In Section 3 we demonstratedthat λ satisfies this constraint if and only if λ belongs to the limit spectrumlim ε σ ( A ε ) = [ θ σ ( A θ ) . For λ in the gaps of this limit spectrum, i.e. when λ satisfies the inequality (cid:12)(cid:12)(cid:12)(cid:12) v ( h ) + ( a v ′ )( h ) − λ v ( h ) Z Y ρ (cid:12)(cid:12)(cid:12)(cid:12) > , the roots µ , µ satisfy | µ | < | µ | >
1. For such λ , we can construct “unstable” solutions, oneof which decays and the other grows. Indeed, denoting by κ and κ the eigenvectors corresponding9o µ and µ respectively, we find in the interval [ ⌈ d + ⌉ ε , ∞ ) that w given by (4.28), (4.29) satisfies w ( y + 1) = µ j w ( y ) if ( l ⌈ d + ⌉ ε , m ⌈ d + ⌉ ε ) = κ j , j = 1 , . Similarly, in the interval ( −∞ , ⌊ d − ⌋ ε ], one has w ( y ) = µ − j w ( y −
1) if ( l ⌊ d − ⌋ ε − , m ⌊ d − ⌋ ε − ) = κ j , j = 1 , . For w to decay to the left and right ofthe defect, we set ( l ⌈ d + ⌉ ε , m ⌈ d + ⌉ ε ) = κ and ( l ⌊ d − ⌋ ε − , m ⌊ d − ⌋ ε − ) = κ . In this way we ensure that w ( y + 1) = µ w ( y ) for y ∈ [ ⌈ d + ⌉ ε , ∞ ) ,w ( y −
1) = µ − w ( y ) = µ w ( y ) for y ∈ ( −∞ , ⌊ d − ⌋ ε ] . (4.30)We extend w into the cells as follows: w ( y ) = µ − w ( y + 1) for y ∈ I r := ( d + ε , ⌈ d + ⌉ ε (cid:1) ,w ( y ) = µ − w ( y −
1) for y ∈ I l := ( ⌊ d − ⌋ ε , d − ε ) . We choose κ and κ so that the constructed w matches the value of u at the ends of D : w ( d + /ε ) = u ( d + ) , w ( d − /ε ) = u ( d − ) . Note that the normalisation factor for κ , κ depends on ε in general, but it is nevertheless boundeduniformly in ε .The second equation and third equations of (4.24) determine w in the stiff component up to anarbitrary additive constant in each interval Y + z , z ∈ I ε , and in the stiff intervals ( I l ∪ I r ) ∩ ε − Ω ε adjacent to D . In the intervals Y + z , z ∈ I ε , we choose this constant so that w ( h + z ) = 0 if ε ( h + z ) ≥ d + ,w (1 + z ) = 0 if ε ( h + z ) ≤ d − . (4.31)In the intervals ( I l ∪ I r ) ∩ ε − Ω ε we choose the value of the constant so that w ( d − /ε ) = w ( d + /ε ) = 0 . (4.32)In the soft component Y + z , z ∈ I ε we do not require w to satisfy any equation. Instead we makea specific choice of w as follows. Let f ∈ C ∞ ( Y ) be a positive function, then we define w ( z + y ) := w ( z ) + c z Z y fa , y ∈ Y , z ∈ I ε , where the coefficients c z are chosen so that w is continuous on R \ D . Thus, we have( a w ′ )( z + ) = ( a w ′ )(( z + h ) − ) = 0 , z ∈ I ε . Finally, conditions (4.31), (4.32) imply that we can extend w by zero into the soft intervals in thecells adjacent to D : w ≡ (cid:2)(cid:0) ⌊ d − ⌋ ε , ⌊ d − ⌋ ε + h (cid:1) ∪ (cid:0) ⌊ d + ⌋ ε , ⌊ d + ⌋ ε + h (cid:1)(cid:3) \ ε − D. It remains to define u on D so that it vanishes on the boundary of D and so that its co-derivativematches the co-derivative of w ( x/ε ) + ε w ( x/ε ) . We require( a D u ′ ) (cid:0) ( d + ) − (cid:1) = J := ( ( a w ′ ) (cid:0) ( d + /ε ) + (cid:1) , if d + /ε ∈ (cid:2) ⌊ d + ⌋ ε , ⌊ d + ⌋ ε + h (cid:1) , ( a w ′ ) (cid:0) ( d + /ε ) + (cid:1) , if d + /ε ∈ (cid:2) ⌊ d + ⌋ ε + h, ⌈ d + ⌉ ε (cid:1) , ( a D u ′ ) (cid:0) ( d − ) + (cid:1) = J := ( ( a w ′ ) (cid:0) ( d − /ε ) − (cid:1) , if d − /ε ∈ (cid:2) ⌊ d − ⌋ ε , ⌊ d − ⌋ ε + h (cid:1) , ( a w ′ ) (cid:0) ( d − /ε ) − (cid:1) , if d − /ε ∈ (cid:2) ⌊ d − ⌋ ε + h, ⌈ d − ⌉ ε (cid:1) . In order to fulfil the above conditions we take a smooth cut-off function χ such that χ ( x ) = 0 , x ≤ d − + δ , χ ( x ) = 1 , x ≥ d + − δ , for a sufficiently small δ >
0, and define u ( x ) := J χ ( x ) Z xd + a − D + J (1 − χ ( x )) Z xd − a − D , x ∈ R . λ ∈ σ (cid:0) A N ,D (cid:1) \ (cid:0)S θ σ ( A θ ) (cid:1) . The construction described above guarantees thatthe function u ε, ap ( x ) := ( u ( x ) + εu ( x ) , x ∈ D,w ( x/ε ) + ε w ( x/ε ) , x ∈ R \ D, (4.33)is continuous and has a continuous co-derivative a εD u ′ ε, ap , implying that u ε, ap belongs to the domainof the operator A εD . Moreover, it is not difficult to see that k w ( · /ε ) k L ρεD ( R \ D ) ≤ ε / || w || L ρ ( R \ ε − D ) , k ( a w ′ ) ′ ( · /ε ) k L ρεD ( R \ D ) + k w ( · /ε ) k L ρεD ( R \ D ) ≤ C k w ( · /ε ) k L ρεD ( R \ D ) , k ( a D u ′ ) ′ k L ρεD ( D ) + k u k L ρεD ( D ) ≤ C, (4.34)for some constant C > e.g. [4]) that for all functions f ∈ dom (cid:0) A εD (cid:1) ⊂ L ρ εD ( R ) such that k f k L ρεD ( R ) = 1 , one hasdist (cid:0) λ , σ ( A εD ) (cid:1) ≤ (cid:13)(cid:13) ( A εD − λ ) f (cid:13)(cid:13) L ρεD ( R ) . Straightforward calculations show that( A εD − λ ) u ε, ap = − ε ( a D u ′ ) ′ ( x ) − ελ ρ D u ( x ) , x ∈ D, − ε ( a w ′ ) ′ ( x/ε ) − ε λ ρ w ( x/ε ) , x ∈ Ω ε \ D, − ε λ ρ w ( x/ε ) , x ∈ Ω ε \ D. Then (4.34) readily implies there exists
C > (cid:13)(cid:13) ( A εD − λ ) u ε, ap (cid:13)(cid:13) L ρεD ( R ) ≤ Cε. (4.35)We establish the following result, which implies Claim 1 of Theorem 2.4. In particular, the secondestimate in (2.6) follows from (4.33), (4.34) and (4.36) below.
Theorem 4.1.
Suppose that λ ∈ σ (cid:0) A N ,D (cid:1) \ (cid:0)S θ σ ( A θ ) (cid:1) .
1. There exists C > , independent of ε , such that dist (cid:0) λ , σ ( A εD ) (cid:1) ≤ C ε.
2. For sufficiently small ε there exist (simple) eigenvalues λ ε of A εD such that | λ ε − λ | ≤ C ε .3. For sufficiently small ε the function u ε, ap is an approximate eigenfunction of A εD : there exists aconstant C > independent of ε such that (cid:13)(cid:13) u ε, ap − u ε (cid:13)(cid:13) L ρεD ( R ) ≤ C ε, (4.36) where u ε is the eigenfunction of A εD corresponding to the eigenvalue λ ε . Proof.
Claim 1 of the theorem follows from (4.35) and the fact that k u ε, ap k L ρεD ( R ) → k u k L ρD ( D ) = 1as ε → , due to (4.34). Claim 2 follows by noting that the essential spectra of A εD and A ε coincide,and that σ ( A ε ) = σ ess ( A ε ) converges to S θ σ ( A θ ), as ε →
0, to which λ does not belong. To proveclaim 3, one can argue as in [20], or [11, Section 11.1]. Namely, it follows from (4.35) and a spectraldecomposition of u ε, ap with respect to the operator A εD that there exists an ε -independent constant C > c εj ∈ R such that (cid:13)(cid:13)(cid:13) u ε, ap − X j ∈ J ε c εj u ε,j (cid:13)(cid:13)(cid:13) L ρεD ( R ) ≤ C ε, where for each ε , J ε := { j : | λ ε,j − λ | ≤ C ε } is a finite set of indices and u ε,j are L ρ εD ( R )-normalisedeigenfunctions of A εD with eigenvalue λ ε,j . Next, the compactness property demonstrated in Theorem5.1, see next section, implies that there is exactly one sequence of simple eigenvalues λ ε convergingto λ , hence (4.36) holds. 11 Spectral completeness of defect eigenvalues and localisa-tion of eigenmodes
The method of asymptotic expansions allows us to show that for any eigenvalue λ of A N ,D , cf. (2.5),in a gap of S θ σ ( A θ ) there exists a sequence of eigenvalues of A εD converging to λ . The conversestatement requires a compactness argument for a corresponding sequence of eigenfunctions of A εD .In this section we use functional analytic techniques, which, unlike Section 6, do not rely on the one-dimensional nature of the problem, to show a decay of the eigenfunctions of A εD outside the defectsufficient to imply the compactness of sequences of eigenfunctions with eigenvalues accumulating inthe gaps of S θ σ ( A θ ). Theorem 5.1.
Let λ ε be an eigenvalue sequence of A εD , u ε be a corresponding sequence of L ( R ) -normalised eigenfunctions, and suppose that λ ε → λ ∈ R \ S θ σ ( A θ ) as ε → . Then λ is aneigenvalue of A N ,D and up to a subsequence u ε → u strongly in L ( R ) , u ε ⇀ u weakly in H ( D ) , where u is an eigenfunction corresponding to λ , extended by zero outside the defect D .Proof. The main ingredient of the proof is demonstrating that the eigenfunction sequences u ε localiseto the defect in the sense that lim ε → || χ ε,α u ε || L ( R ) = 0 ∀ α ∈ (0 , , (5.37)where χ ε,α : C ∞ ( R ) → [0 , , ε >
0, is any smooth cut-off function such that χ ε,α = ( D, −∞ , d − − ε α ] ∪ [ d + + ε α , ∞ ) . Additionally, χ ε,α is constant on each connected component of Ω ε and satisfies the bound sup ε ε α k χ ′ ε,α k L ∞ ( R ) < ∞ . The assertion (5.37) is an immediate consequence of the following lemma, that we demonstratebelow. Lemma 5.2.
Consider a sequence λ ε ∈ [0 , ∞ ) , u ε ∈ L ( R ) , || u ε || L ( R ) = 1 , such that A εD u ε = λ ε ρ εD u ε . If the convergence λ ε → λ ∈ R \ S θ σ ( A θ ) holds as ε → , then there exist sequences v ε , w ε ∈ H ( R ) such that u ε = v ε + w ε with the following properties:1) One has v ′ ε ≡ on Ω ε \ D ;
2) The sequence v ε is localised to defect in the sense of (5.37) ;3) There exists a constant C > such that k w ε k L ( R ) ≤ Cε , k w ′ ε k L ( R ) ≤ Cε. (5.38)Let us prove that λ ∈ σ ( A N ,D ) under the assumption that Lemma 5.2 holds. By substituting ϕ = u ε in the eigenvalue problem for the operator A εD ( cf. (2.4)) Z D a εD u ′ ε ϕ ′ = λ ε Z R ρ εD u ε ϕ ∀ ϕ ∈ H ( R ) , and utilising the boundedness of λ ε , the uniform positivity and boundedness of a j , ρ j , j = 1 , , a D and ρ D , we establish the estimatessup ε || u ε || H ( D ) < ∞ , sup ε k u ′ ε k L (Ω ε \ D ) < ∞ , sup ε k εu ′ ε k L (Ω ε \ D ) < ∞ . (5.39)By (5.39), it is clear that a subsequence of u ε converges weakly in H ( D ). Now, by Lemma 5.2 andthe identity u ε = χ ε,α v ε + (1 − χ ε,α ) v ε + w ε we find that u ε strongly converges to zero in L ( R \ D ).Therefore, there exists u ∈ L ( R ), u ≡ R \ D , such that up to a subsequence u ε → u strongly in L ( R ) , u ε ⇀ u weakly in H ( D ) . a ( · ε ) u ′ ε → L (Ω ε \ D ) . Therefore, for fixed ϕ ∈ H ( R ), we can pass to the limit in (2.4), recalling the identity Z R a εD u ′ ε ϕ ′ = Z D a D u ′ ε ϕ ′ + Z Ω ε \ D a ( xε ) u ′ ε ϕ ′ + Z Ω ε \ D ε a ( xε ) u ′ ε ϕ ′ (5.40)to find that Z D a D u ′ ϕ ′ = λ Z D ρ D u ϕ. (5.41)Finally, by the arbitrariness of ϕ deduce that λ ∈ σ ( A N ,D ) and u is the corresponding eigenfunc-tion. Corollary 5.3.
Claim 2 of Theorem 2.4 holds.
We now prove Lemma 5.2.
Proof of Lemma 5.2.
We start by constructing the representation u ε as the sum of v ε and w ε asfollows. On the defect D = ( d − , d + ), we set v ε = u ε . On each connected component of Ω ε , exceptfor the intervals adjacent to the defect, we define v ε as v ε ( x ) := 1 ε (1 − h ) Z ε ( Y + z ) u ε , x ∈ ε ( Y + z ) , z ∈ I ε , (5.42)where I ε is defined by (4.25). If necessary, we extend v ε continuously by constant from D intothe stiff region adjacent to the defect, i.e. Ω ε ∩ (cid:0)(cid:0) min (cid:8) d − , ε ( ⌊ d − ⌋ ε + h ) (cid:9) , d − (cid:3) ∪ (cid:2) max (cid:8) ε ( ⌊ d + ⌋ ε + h ) , d + (cid:9) , ⌈ d + ⌉ ε (cid:1)(cid:1) . Thus v ε is defined everywhere except the soft component Ω ε \ D , and is piecewiseconstant on the stiff component Ω ε \ D . To define v ε on Ω ε \ D we ensure that the difference w ε := u ε − v ε , x ∈ Ω ε \ D, is extended into the soft component Ω ε \ D so that w ε ∈ H ( R ) and satisfies (cid:0) a ( · ε ) w ′ ε (cid:1) ′ = 0 , on Ω ε \ D. (5.43)Thus we have u ε = v ε + w ε , where v ε , w ε ∈ H ( R ) with v ′ ≡ ε \ D and w ε ≡ D .We first prove (5.38). By construction, for each z ∈ I ε , the function w ε has zero mean value onthe interval ε ( Y + z ) and it is clear, for example by an application of the fundamental theorem ofcalculus, that for each z ∈ I ε one has (cid:12)(cid:12) w ε ( x ) (cid:12)(cid:12) ≤ ε (1 − h ) Z ε ( Y + z ) | w ′ ε | , x ∈ ε ( Y + z ) . (5.44)A version of the same argument implies that since w ε ≡ d − , d + ), on I ε := ( ε ⌊ d − ⌋ ε , ε ⌈ d + ⌉ ε ) wehave (cid:12)(cid:12) w ε ( x ) (cid:12)(cid:12) ≤ ε Z I ε \ D | w ′ ε | , x ∈ I ε . (5.45)Moreover, if the soft component touches the defect on the right, i.e. if d + < ε ( ⌊ d + ⌋ ε + h ) then | w ε ( x ) | ≤ ( ε ⌊ d + ⌋ ε + εh − d + ) − Z ( d + ,ε ⌊ d + ⌋ ε + εh ) | w ′ ε | , x ∈ ( d + , ε ⌊ d + ⌋ ε + εh ) (5.46)and if the soft component touches the defect on the left, i.e. d − ≤ ε ( ⌊ d − ⌋ ε + h ), then | w ε ( x ) | ≤ ( d − − ε ⌊ d − ⌋ ε ) − Z ( ε ⌊ d − ⌋ ε ,d − ) | w ′ ε | , x ∈ ( ε ⌊ d − ⌋ ε , d − ) . (5.47)13n the soft component Ω ε \ I ε = S z ∈I ε ε ( Y + z ), we note that since w ε solves (5.43), the maximumprinciple implies sup ( εz,ε ( z + h )) | w ε | = max (cid:8)(cid:12)(cid:12) w ε ( εz ) (cid:12)(cid:12) , (cid:12)(cid:12) w ε ( ε ( z + h )) (cid:12)(cid:12)(cid:9) ∀ z ∈ I ε . This fact, along with inequalities (5.44) and (5.45), implies that k w ε k L ( ε ( Y + z )) ≤ ε max (cid:8) k w ′ ε k L ( ε ( Y + z )) , k w ′ ε k L ( ε ( Y + z − (cid:9) . Putting the above inequalities together, it follows that k w ε k L ( R ) = Z I ε | w ε | + X z ∈I ε Z ε ( Y + z ) | w ε | + X z ∈I ε Z ε ( Y + z ) | w ε | ≤ ε k w ′ ε k L (Ω ε \ D ) . (5.48)Straightforward calculations show that due to (5.43) we have on the soft componentsup ε ( Y + z ) | w ′ ε | ≤ ( εh ) − || a || L ∞ ( Y ) || a − || L ∞ ( Y ) (cid:16)(cid:12)(cid:12) w ε ( εz ) (cid:12)(cid:12) + (cid:12)(cid:12) w ε ( εz + εh ) (cid:12)(cid:12)(cid:17) . Similarly, if the soft component touches the defect on the right, i.e. if d + < ε ( ⌊ d + ⌋ ε + h ) thensup ( d + ,ε ⌊ d + ⌋ ε + εh ) | w ′ ε | ≤ ( ε ⌊ d + ⌋ ε + εh − d + ) − || a || L ∞ ( Y ) || a − || L ∞ ( Y ) (cid:12)(cid:12) w ε ( ε ⌊ d + ⌋ ε + εh ) (cid:12)(cid:12) , and if the soft component touches the defect on the left, i.e. d − ≤ ε ( ⌊ d − ⌋ ε + h ), thensup ( ε ⌊ d − ⌋ ε ,d − ) | w ′ ε | ≤ ( d − − ε ⌊ d − ⌋ ε ) − || a || L ∞ ( Y ) || a − || L ∞ ( Y ) (cid:12)(cid:12) w ε ( ε ⌊ d − ⌋ ε ) (cid:12)(cid:12) . Consequently, from (5.44)–(5.47) and the above assertions, we obtain k w ′ ε k L (Ω ε \ D ) ≤ C k w ′ ε k L (Ω ε \ D ) . (5.49)It remains to bound w ′ ε on the stiff component Ω ε \ D , which in combination with (5.48) and(5.49) yields the estimates (5.38). To this end, note that by setting ϕ = w ε in (2.4), using theidentity u ε = v ε + w ε and the facts that v ′ ε = 0 in Ω ε \ D and w ε ≡ D , we have Z Ω ε \ D ε a ( · ε ) u ′ ε w ′ ε + Z Ω ε \ D a ( · ε ) | w ′ ε | dx = λ ε Z Ω ε \ D ρ ( · ε ) u ε w ε + λ ε Z Ω ε \ D ρ ( · ε ) u ε w ε . Hence, by the H¨older inequality we deduce that || w ′ ε || L (Ω ε \ D ) ≤ ε || εu ′ ε || L (Ω ε \ D ) || w ′ ε || L (Ω ε \ D ) + C (cid:0) || u ε || L ( R ) || w ε || L ( R ) (cid:1) for some C > , and utilising (5.39), (5.48), (5.49) yields k w ′ ε k L (Ω ε \ D ) ≤ Cε. (5.50)Hence, by (5.48), (5.49), (5.50) and the fact w ≡ D , it follows that (5.38) holds.We now prove Claim 2. For a fixed ϕ ∈ H ( R ) we take a test function χ ε,α ϕ in (2.4), use theidentity u ′ ε ( χ ε,α ϕ ) ′ = ( u ε χ ε,α ) ′ ϕ ′ − u ε χ ′ ε,α ϕ ′ + u ′ ε χ ′ ε,α ϕ and the decomposition u ε = v ε + w ε to arriveat the equation Z R (cid:0) a εD ( χ ε,α v ε ) ′ ϕ ′ − λ ε ρ εD χ ε,α v ε ϕ (cid:1) = Z R (cid:0) λ ε ρ εD w ε χ ε,α ϕ − a εD ( w ε χ ε,α ) ′ ϕ ′ + a εD χ ′ ε,α ( u ε ϕ ′ − u ′ ε ϕ ) (cid:1) . By inequalities (5.38), (5.39), the fact that χ ′ ε,α ≡ ε , and sup ε ε α | χ ′ ε,α | < ∞ we can estimatethe right-hand side as follows: (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:0) λ ε ρ εD w ε χ ε,α ϕ − a εD ( w ε χ ε,α ) ′ ϕ ′ + a εD χ ′ ε,α ( u ε ϕ ′ − u ′ ε ϕ ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε − α || ϕ || H ( R ) . ε → sup ϕ ∈ H ( R ) || ϕ || H R ) =1 (cid:12)(cid:12)(cid:12)(cid:12)Z R (cid:0) a εD ( χ ε,α v ε ) ′ ϕ ′ − λ ε ρ εD χ ε,α v ε ϕ (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , Notice that ( χ ε,α v ε ) ′ ≡ ε and D , and therefore Z R (cid:0) a εD ( χ ε,α v ε ) ′ ϕ ′ − λ ε ρ εD χ ε,α v ε ϕ (cid:1) = X z ∈ Z Z z + hz a (cid:0) R ε ( χ ε,α v ε ) (cid:1) ′ (cid:0) R ε ( ϕ ) (cid:1) ′ − λ ε Z R ρ R ε ( χ ε,α v ε ) R ε ( ϕ ) , where R ε : L ρ ε ( R ) → L ρ ( R ) is the unitary transformation R ε ( f )( y ) = ε / f ( εy ). It follows that for z ε := R ε ( χ ε,α v ε ) one has lim ε → sup ϕ ∈ H ( R ) || ϕ || H R ) =1 (cid:12)(cid:12)(cid:12)(cid:12)Z Ω a z ′ ε ϕ ′ − λ ε Z R ρz ε ϕ (cid:12)(cid:12)(cid:12)(cid:12) = 0 , (5.51)where Ω := S z ∈ Z ( Y + z ).We now argue as in the demonstration of a Weyl’s criterion for quadratic forms, see [14, Ap-pendix], to show the above condition implies that z ε necessarily converges strongly to zero in L ( R ).Taking test functions in (5.51) from H + = { v ∈ H ( R ) : v ′ ≡ R \ Ω } , we see that the mapping F ε : H + → R given by F ε ( v ) := Z Ω a z ′ ε v ′ − λ ε Z R ρz ε v, v ∈ H + , (5.52)is linear and continuous, i.e. F ε belongs to H − , the space of bounded linear functionals on H + ,with lim ε → || F ε || H − = 0 . (5.53)In Appendix B below, we use standard arguments to demonstrate that there is a unitary map Ψ ◦ U and an element f ε of the space h − := (cid:8) f : (0 , π ) → ℓ measurable : (cid:0) λ n ( θ ) + 1 (cid:1) − / f ( θ, n ) ∈ L (0 , π ; ℓ ) (cid:9) , (5.54)such that H − h F ε , v i H + = X n ∈ N Z π f ε ( θ, n )(Ψ U ) v ( θ, n ) d θ ∀ v ∈ H + , || F ε || H − = vuutX n ∈ N Z π (cid:12)(cid:12) f ε ( θ, n ) (cid:12)(cid:12) λ n ( θ ) + 1 d θ. (5.55)(We recall that λ n ( θ ) are the eigenvalues of the operator A θ , see Section 2). Now, by applying thetransform Ψ U to (5.52), we find that H − h F ε , v i H + = X n ∈ N Z π (cid:0) λ n ( θ ) − λ ε (cid:1) (Ψ U ) z ε ( θ, n )(Ψ U ) v ( θ, n ) d θ. This equality, the formulae (5.55) and the fact that Ψ U unitarily maps H − to h − implies that f ε ( θ, n ) = ( λ n ( θ ) − λ ε )(Ψ U ) z ε ( θ, n )almost everywhere in θ, and || F ε || H − = X n ∈ N Z π (cid:0) λ n ( θ ) − λ ε (cid:1) λ n ( θ ) + 1 (cid:12)(cid:12) (Ψ U ) z ε ( θ, n ) (cid:12)(cid:12) d θ.
15y assumption, one has λ ε → λ / ∈ S θ σ ( A θ ) = P n ∈ N (cid:2) min θ λ n ( θ ) , max θ λ n ( θ ) (cid:3) , and therefore there existsa constant c > ε the inequality (cid:12)(cid:12) λ n ( θ ) − λ ε (cid:12)(cid:12) > c holds for all n ∈ N and all θ ∈ [0 , π ). Hence, the above equality and (5.53) imply thatlim ε → X n ∈ N Z π (cid:12)(cid:12) (Ψ U ) z ε ( θ, n ) (cid:12)(cid:12) d θ ≤ c lim ε → || F ε || H − = 0 . Finally, since (Ψ U ) z ε = (Ψ UR ε )( χ ε,α v ε ), and Ψ UR ε is unitary, it follows that Claim 2 holds. In Section 5 we demonstrate that for eigenvalue sequences converging to a point in a gap in the limitspectrum S θ σ ( A θ ), the corresponding eigenfunctions u ε converge to zero in L outside the defect D , as ε → α ∈ (0 , . In this section, using the fact that one-dimensional problems admit anexplicit form of solutions in terms of the fundamental system and employing standard techniquesfrom the theory of ordinary differential equations, we provide a stronger statement on the rateof decay outside the defect. Namely, we show that the eigenfunctions u ε decay at an acceleratedexponential rate outside of the defect, which is Theorem 2.4, Claim 3.As in Section 5, we assume a sequence of eigenvalues λ ε of A εD converges to λ ∈ R \ S θ σ ( A θ ) as ε →
0, and consider the corresponding sequence u ε of L ( R )-normalised eigenfunctions, i.e. Z R a εD u ′ ε ϕ ′ = λ ε Z R ρ εD u ε ϕ, ∀ ϕ ∈ H ( R ) . Recalling the unitary operator R ε : L ρ ε ( R ) → L ρ ( R ) given by R ε ( f )( y ) = ε / f ( εy ), we notethat for all z ∈ I ε (see (4.25)), the function ˜ u ε := R u ε solves − ( a ˜ u ′ ε ) ′ = λ ε ρ ˜ u ε on Y + z, (6.56) − ε − ( a ˜ u ′ ε ) ′ = λ ε ρ ˜ u ε on Y + z, (6.57)and satisfies the interface conditions˜ u ε | Y + z ( z + h ) = ˜ u ε | Y + z ( z + h ) , ( a ˜ u ′ ε ) (cid:0) ( z + h ) − (cid:1) = ε − ( a ˜ u ′ ε ) (cid:0) ( z + h ) + (cid:1) , ˜ u ε | Y + z +1 ( z + 1) = ˜ u ε | Y + z ( z + 1) , ( a ˜ u ′ ε ) (cid:0) ( z + 1) + (cid:1) = ε − ( a ˜ u ′ ε ) (cid:0) ( z + 1) − (cid:1) . (6.58)There exist solutions v ε , v ε to the equation − ( a u ′ ) ′ = λ ε ρ u, on Y , and solutions w ε , w ε to theequation − ε − ( a u ′ ) ′ = λ ε ρ u, on Y , such that (cid:18) v ε v ε a v ε ′ a v ε ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) y =0 = (cid:18) (cid:19) , (cid:18) w ε w ε a w ε ′ a w ε ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) y = h = (cid:18) (cid:19) . The solution ˜ u ε to (6.56), (6.57), z ∈ I ε , admits the representation˜ u ε ( y ) = ( a εz v ε ( y − z ) + b εz v ε ( y − z ) , y ∈ Y + z,c εz w ε ( y − z ) + d εz w ε ( y − z ) , y ∈ Y + z. (6.59)For all ε, the coefficients a εz , b εz c εz and d εz , z ∈ I ε , are related to each other by the conditions (6.58),as follows: c εz = a εz v ε ( h ) + b εz v ε ( h ) , ε − d εz = a εz ( a v ε ′ )( h ) + b εz ( a v ε ′ )( h ) ,a εz +1 = c εz w ε (1) + d εz w ε (1) , ε b εz +1 = c εz ( a w ε ′ )(1) + d εz ( a w ε ′ )(1) . Eliminating c εz and d εz gives the iterative system (cid:18) a εz +1 b εz +1 (cid:19) = M ε (cid:18) a εz b εz (cid:19) , (6.60)16here the matrix M ε is given by M ε = v ε ( h ) w ε (1) + ε ( a v ε ′ )( h ) w ε (1) v ε ( h ) w ε (1) + ε ( a v ε ′ )( h ) w ε (1) ε − v ε ( h )( a w ε ′ )(1) + ( a v ε ′ )( h )( a w ε ′ )(1) ε − v ε ( h )( a w ε ′ )(1) + ( a v ε ′ )( h )( a w ε ′ )(1) ! . (6.61)It follows from the property that the modified Wronskian is constant,det (cid:18) v ε v ε a v ε ′ a v ε ′ (cid:19) ≡ , det (cid:18) w ε w ε a w ε ′ a w ε ′ (cid:19) ≡ , that the characteristic polynomial of M ε is given bydet( M ε − µI ) = µ − µh ε + 1 ,h ε = v ε ( h ) w ε (1) + ε ( a v ε ′ )( h ) w ε (1)+ ε − v ε ( h )( a w ε ′ )(1) + ( a v ε ′ )( h )( a w ε ′ )(1) . (6.62)Recalling, from Section 3.1, the fundamental solutions v , v of ( cf. (3.9), (3.11)) − ( a u ′ ) ′ = λ ρ u in Y , satisfying v (0) v (0)( a v ′ )(0) ( a v ′ )(0) ! = ! , we shall prove in the second half of this section the following property. Lemma 6.1.
The following convergence holds: lim ε → h ε = v ( h ) + ( a v ′ )( h ) − λ v ( h ) Z Y ρ . (6.63)Assuming that (6.63) holds, since λ ∈ R \ S θ σ ( A θ ) , or equivalently (see Section 3.1) λ is suchthat ( cf. (3.15)) (cid:12)(cid:12)(cid:12)(cid:12) v ( h ) + ( a v ′ )( h ) − λ v ( h ) Z Y ρ (cid:12)(cid:12)(cid:12)(cid:12) > , for sufficiently small ε we find that | h ε | > µ ε , µ ε of the matrix M ε satisfy the identity µ ε µ ε = 1and the nature of ˜ u ε away from the defect is determined by the coefficient h ε . In particular, if | h ε | > µ ε , µ ε are such that | µ ε | < | µ ε | > v g , v d on R \ (cid:0) −⌊ d − ⌋ ε , ⌈ d + ⌉ ε (cid:1) that grow and decay respectively. In this case, for u ε be anelement of L ( R ) it is necessary that u ε is proportional to the decaying solution v d , which takes theform v d ( x ) = exp (cid:16) ln | µ ε | ε dist( x, D ) (cid:17) p ε ( x/ε ) , x ∈ [ d + , ∞ ) , exp (cid:16) ln | µ ε | ε dist( x, D ) (cid:17) p ε ( x/ε ) , x ∈ ( −∞ , d − ] , for some periodic (respectively, anti-periodic) functions p ε , p ε , when h ε > h ε < ν satisfying ν < − ln | µ ε | = (cid:12)(cid:12) ln | µ ε | (cid:12)(cid:12) the product g ν/ε u ε is in L (Ω),where g ν/ε is defined by (2.7). Then the third claim of Theorem 2.4 follows by noticing that by(6.63) µ ε converges to µ , the smallest root of µ − hµ + 1 , where h := v ( h ) + ( a v ′ )( h ) − λ v ( h ) Z Y ρ , as ε → Proof of Lemma 6.1. η εj := v εj − v j a v εj ′ − a v j ′ ! , j = 1 , , (6.64)solves the initial-value problem η εj ′ = Φ ε η εj + Ψ εj in Y , η εj (0) = 0 , j = 1 , , (6.65)for the matrix Φ ε and vector Ψ εj , j = 1 , , given byΦ ε = a − − λ ε ρ ! , Ψ εj = λ − λ ε ) ρ v j ! , j = 1 , . Since λ ε → λ the solutions to (6.65) converge uniformly on Y to the trivial solution of η ′ = Φ η in Y , η (0) = 0 , where Φ is the limit of Φ ε , as ε → e.g. [18, Theorem 1.6.1]). Namely, we have (cid:12)(cid:12) η εj ( y ) (cid:12)(cid:12) = | η εj ( y ) − η ( y ) | ≤ C | λ ε − λ | , j = 1 , , for some constant C independent of ε . In particular, recalling (6.64), it follows thatlim ε → v εj ( h ) = v j ( h ) , lim ε → ( a v εj ′ )( h ) = ( a v ′ j )( h ) , j = 1 , . (6.66)Similarly, it is easy to see that w εj and a w εj ′ converge uniformly on Y to w j and a w ′ j , where w j , j = 1 , a w ′ ) ′ = 0 satisfying w ( h ) w ( h )( a w ′ )( h ) ( a w ′ )( h ) ! = ! . Since w ≡ a w ′ ≡ Y we see that w ε w ε ( a w ε ′ ) ( a w ε ) ′ ! → R yh a − ! uniformly on Y as ε → . (6.67)Furthermore, by the fundamental theorem of calculus and the fact − ε − ( a w ε ′ ) ′ = λ ε ρ w ε , wehave ε − ( a w ε ) ′ (1) − ε − ( a w ε ) ′ ( h ) = − λ ε Z h ρ w ε , and since Z h ρ w ε − w ε ( h ) Z h ρ = Z h ρ (cid:0) w ε − w ε ( h ) (cid:1) = Z Y ρ ( y ) (cid:18)Z yh w ε ′ (cid:19) d y, it follows that (cid:12)(cid:12)(cid:12)(cid:12) ε − ( a w ε ′ )(1 − ) − ε − ( a w ε ′ )( h + ) + λ ε w ε ( h ) Z h ρ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ ε Z Y ρ ( y ) (cid:18)Z yh w ε ′ (cid:19) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ | λ ε ||| ρ || L ∞ || w ε ′ || L ∞ , which together with (6.67) implieslim ε → (cid:12)(cid:12)(cid:12)(cid:12) ε − ( a w ε ′ )(1) − ε − ( a w ε ′ )( h ) + w ε ( h ) λ ε Z h ρ (cid:12)(cid:12)(cid:12)(cid:12) = 0 . Therefore lim ε → ε − (cid:0) a w ε ′ (cid:1) (1) = − λ Z Y ρ . (6.68)Finally, assertions (6.66), (6.67) and (6.68) imply (6.63), as required.18 Resolvent estimates for the problem without defect
In this section we study the behaviour of the unperturbed periodic operator A ε in the operatornorm as ε →
0. In particular, we construct a full asymptotic expansions for the resolvent of A ε using a version of the asymptotic framework developed in [7], see Theorem 7.2 below. This directlyimplies the order-sharp operator norm resolvent convergence estimate, uniform in θ , formulated inTheorem 2.2. The latter, in turn, implies the uniform in θ convergence, as ε →
0, of the spectralband functions λ εn ( θ ) to λ n ( θ ), n ∈ N , which is also order-sharp.Recall the operator A ε in L ρ ε ( R ) associated with the bilinear form β ε ( u, v ) = Z Ω ε a ( · ε ) u ′ v ′ + Z Ω ε ε a ( · ε ) u ′ v ′ , u, v ∈ H ( R ) . By a scaled version of the Floquet-Bloch transform which is given as the continuous extension ofthe following action on e.g. continuous functions with compact support( U ε f )( θ, y ) = r ε π X z ∈ Z f (cid:0) ε ( y − z ) (cid:1) e i θz , y ∈ Y, θ ∈ [0 , π ) , (7.69)we see that U ε unitarily maps L ρ ε ( R ) to the Bochner space L (cid:0) , π ; L ρ ( Y ) (cid:1) and U ε A ε f ( θ, · ) = A εθ U ε f ( θ, · ). Here, A εθ is the operator defined in L ρ ( Y ) and associated with the form β ε ( u, v ) := Z Y a u ′ v ′ + ε − Z Y a u ′ v ′ , u, v ∈ H θ ( Y ) . We recall that H θ ( Y ) is the complex Hilbert space of H ( Y )-functions that are θ -quasiperiodic. Weequip the space H θ ( Y ) with the graph norm ||| u ||| := sZ Y a | u ′ | + Z Y a | u ′ | + Z Y ρ | u | , (7.70)and consider the subspace V θ := (cid:8) v ∈ H θ ( Y ) : v ′ ≡ Y (cid:9) and its orthogonal complement V ⊥ θ in H θ with respect to the inner product associated with ||| · ||| .The following result, established in [8], is of fundamental importance in studying the asymptotics of A ε , equivalently A εθ . Lemma 7.1.
There exists a constant C P > , independent of θ , such that ||| P ⊥ θ u ||| ≤ C P ||√ a u ′ || L ( Y ) , ∀ u ∈ H θ ( Y ) , (7.71) where P ⊥ θ is the orthogonal projection of H θ ( Y ) onto V ⊥ θ . For θ ∈ [0 , π ) and all f ∈ L ρ ( Y ) , we consider the resolvent problem − (cid:0) ( ε − a + a ) u εθ ′ (cid:1) ′ + ρu εθ = ρf on (0 , . (7.72)We look for an asymptotic expansion of u εθ in the form u εθ = ∞ X n =0 ε n u (2 n ) θ , u (2 n ) θ ∈ H θ ( Y ) ∀ n ∈ N . (7.73)The following result holds. See Appendix A below for further information on the Floquet-Bloch transform. heorem 7.2. For each θ ∈ [0 , π ) and f ∈ L ρ ( Y ) , consider the unique solution u (0) θ ∈ V θ to theproblem Z Y a ( u (0) θ ) ′ ϕ ′ + Z Y ρu (0) θ ϕ = Z Y ρf ϕ ∀ ϕ ∈ V θ , and for all n ∈ N consider the unique solution u (2 n ) θ ∈ V ⊥ θ to − (cid:16) a (cid:0) u (2 n ) θ (cid:1) ′ (cid:17) ′ = (cid:16) a (cid:0) u (2( n − θ (cid:1) ′ (cid:17) ′ − ρu (2( n − θ + δ n ρf, where δ n is the Kronecker delta function. Then, for each N ∈ N the sum U ( N ) θ := N X n =0 ε n u (2 n ) θ approximates the solution u εθ to (7.72) in the following sense: ||| u εθ − U ( N ) θ ||| ≤ C N +1) P ε N +1) (cid:13)(cid:13) f (cid:13)(cid:13) L ρ ( Y ) . Remark 7.3.
By an application of the min-max principle, Theorem 7.2 implies that the n -th eigen-value λ εn ( θ ) of the operator A εθ is ε -close, uniformly in θ, to the n -th eigenvalue λ n ( θ ) of A θ , i.e.for each n ∈ N there exists a constant c n > such that (cid:12)(cid:12) λ εn ( θ ) − λ n ( θ ) (cid:12)(cid:12) ≤ c n ε ∀ θ ∈ [0 , π ) . In particular, this indirectly implies, since λ n is the uniform limit of continuous functions, that λ n is continuous in θ . A direct proof of this fact can be arrived at by the definition of the operators A θ and the continuity properties (in the Hausdorff sense) of their domains D ( A θ ) , see [8, Appendix B].Proof. Substituting (7.73) into (7.72) and equating powers of ε yields a system of recurrence relationsfor the functions u (2 n ) θ , n ∈ N . The first equation in this system, which corresponds to ε − , is − (cid:16) a (cid:0) u (0) θ (cid:1) ′ (cid:17) ′ = 0 on (0 , , (7.74)which implies that u (0) θ ∈ V θ = (cid:8) v ∈ H θ ( Y ) : v ′ ≡ Y (cid:9) (recall that a ≡ Y ). Theremaining equations, obtained by considering the terms of order ε j , j = 0 , , , ... are − (cid:16) a (cid:0) u (2 n ) θ (cid:1) ′ (cid:17) ′ = (cid:16) a (cid:0) u (2( n − θ (cid:1) ′ (cid:17) ′ − ρu (2( n − θ + δ n ρf, on (0 , , n ∈ N , (7.75)where, as before, δ in denotes the Kronecker delta function. The existence of solutions to differentialequations with degenerate coefficients such as (7.75) was first studied in [10] for the case θ = 0, andit was shown therein that existence is guaranteed by inequalities of the type (7.71). By followingthis general framework, and it can be readily shown that (7.71) implies the following result. Lemma 7.4.
For a given F ∈ H − θ ( Y ) , the dual space of H θ ( Y ) , there exist (infinitely many)solutions u to the problem Z Y a u ′ ϕ ′ = H − θ ( Y ) h F, ϕ i H θ ( Y ) ∀ ϕ ∈ H θ ( Y ) , if and only if F satisfies the condition H − θ ( Y ) h F, v i H θ ( Y ) = 0 ∀ v ∈ V θ . Such solutions are unique in V ⊥ θ , i.e. for any two solutions u , u one has u − u ∈ V θ . Z Y ( a u (2 n ) θ ) ′ ϕ ′ + Z Y ρu (2 n ) θ ϕ = δ n Z Y ρf ϕ ∀ ϕ ∈ V θ , n ∈ N , (7.76)hold. The equation for n = 0 uniquely determines u (0) θ and for n ≥
1, due to the choice (7.70) ofthe norm on H θ ( Y ), demonstrates that u (2 n ) θ ∈ V ⊥ θ . Substituting ϕ = u (0) θ into the identity (7.76)for n = 0, recalling (7.70), the fact that a ( u (0) θ ) ′ ≡ ||| u (0) θ ||| ≤ k f k L ρ ( Y ) (cid:13)(cid:13) u (0) θ (cid:13)(cid:13) L ρ ( Y ) . Hence, u (0) θ satisfies the bound ||| u (0) θ ||| ≤ k f k L ρ ( Y ) ∀ θ ∈ [0 , π ) . (7.77)By Lemmas 7.1 and 7.4, the solution u (2 n ) θ ∈ V ⊥ θ to (7.75) is unique and ||| u (2 n ) θ ||| ≤ C P (cid:13)(cid:13) √ a u (2 n ) θ (cid:13)(cid:13) L ( Y ) . (7.78)Equations (7.75) and the orthogonality of V θ and V ⊥ θ with respect to the norm (7.70), in particular,the orthogonality of u (0) θ and u (2) θ , imply that Z Y a (cid:12)(cid:12)(cid:12)(cid:0) u (2 n ) θ (cid:1) ′ (cid:12)(cid:12)(cid:12) = δ n Z Y ρf u (2 n ) θ − (1 − δ n ) (cid:18)Z Y a (cid:0) u (2( n − θ (cid:1) ′ (cid:0) u (2 n ) θ (cid:1) ′ + Z Y ρu (2( n − θ u (2 n ) θ (cid:19) , n ≥ , and (7.78) yields ||| u (2) θ ||| ≤ C P (cid:13)(cid:13) f (cid:13)(cid:13) L ρ ( Y ) , ||| u (2 n ) θ ||| ≤ C P ||| u (2( n − ||| , n ≥ . By iterating the above inequalities we establish that ||| u (2 n ) θ ||| ≤ C nP (cid:13)(cid:13) f (cid:13)(cid:13) L ρ ( Y ) , n ≥ . (7.79)Having identified each term in the expansion, for each n ∈ N we define the remainder R εθ (drop-ping the index N for brevity), according to the formula u εθ = N X n =0 ε n u (2 n ) θ + ε N R εθ , (7.80)and find, via (7.74) and (7.75), that R εθ ∈ H θ ( Y ) solves the problem − (cid:0) ( ε − a + a )( R εθ ) ′ (cid:1) ′ + ρR εθ = δ N ρf + (cid:0) a ( u (2 N ) θ ) ′ (cid:1) ′ − ρu (2 N ) θ on (0 , , that is Z Y ε − a ( R εθ ) ′ v ′ + Z Y a ( R εθ ) ′ v ′ + Z Y ρR εθ v = δ N Z Y ρf v − Z Y a ( u (2 N ) θ ) ′ v ′ − Z Y ρu (2 N ) θ v, ∀ v ∈ H θ ( Y ) . Setting v ∈ V θ , recalling the norm (7.70) and (7.76), demonstrates that R εθ ∈ V ⊥ θ . Additionally,setting v = R εθ above implies that ε − Z Y a (cid:12)(cid:12) ( R εθ ) ′ (cid:12)(cid:12) ≤ δ N Z Y ρf R εθ − Z Y a ( u (2 N ) θ ) ′ ( R εθ ) ′ − Z Y ρu (2 N ) θ R εθ , ||| R εθ ||| ≤ C N +1) P ε k f k L ρ ( Y ) . Finally, by combining this inequality with (7.80) we deduce that ||| u εθ − N X n =0 ε n u (2 n ) θ ||| ≤ C N +1) P ε N +1) k f k L ρ ( Y ) , as required. A Appendix: Norm-resolvent convergence of A ε and the limitoperator A H to be the closure in L ρ ( R ) of ( cf. the end of Section 5) H + = (cid:8) v ∈ H ( R ) : v ′ ≡ := [ z ∈ Z ( Y + z ) (cid:9) . Both H and H + are Hilbert spaces when equipped with the inner products inherited from L ρ ( R )and H ( R ) respectively, and clearly H + is densely defined in H with continuous embedding (recall ρ is taken to be uniformly positive and bounded). The norm of H + , which is the standard H -norm,is equivalent to the graph norm || · || H + := (cid:16) || · || L ρ ( R ) + β ( · , · ) (cid:17) / , (1.81)where β is the bilinear form β ( u, v ) := Z Ω a u ′ v ′ , u, v ∈ H + . We shall henceforth consider H + to be equipped with the graph norm (1.81), and denote by H − the dual space consisting of bounded linear functionals on H + . As β is a non-negative closedsymmetric quadratic form it generates a densely defined non-negative self-adjoint linear operator A . The domain D ( A ) is the dense subset of H + consisting of the solutions to the problem: foreach f ∈ H we consider u ∈ H + the unique solution to the problem β ( u, v ) + Z R ρuv = Z R ρf v ∀ v ∈ H + , and set A u = f − u for u ∈ D ( A ). The operator A is unitarily equivalent to the fibre integraloperator R θ A θ , cf. Remark 2.1, and the unitary map is given by the continuous extension of theFloquet-Bloch transform U , cf. [15, Section 2.2] which acts on smooth functions f with compactsupport as U f ( θ, y ) := 1 √ π X z ∈ Z f ( y − z ) e iθz , θ ∈ [0 , π ) , y ∈ Y .
Indeed, U is well-known to be a unitary operator between L ρ ( R ) and the Bochner space L (cid:0) , π ; L ρ ( Y ) (cid:1) and it is straightforward to see that U A f ( θ ; · ) = A θ U f ( θ ; · ) , ∀ f ∈ L ρ ( R ) , θ ∈ [0 , π ) . U unitarily maps H + to the space L (0 , π ; V θ ) (we recall that V θ = (cid:8) v ∈ H θ ( Y ) : v ′ ≡ Y (cid:9) ). It is easy to verify that the spectrum of A coincides with the unionof the spectra of A θ over all θ ∈ [0 , π ), i.e. σ ( A ) = [ θ σ ( A θ ) = [ n ∈ N (cid:2) min θ λ n ( θ ) , max θ λ n ( θ ) (cid:3) . Theorem 2.2 implies in particular that A ε converges at the rate ε in the norm-resolvent sense to A , i.e. there exists a constant C > (cid:13)(cid:13) R ε ( A ε + 1) − R − ε − ( A + 1) − (cid:13)(cid:13) L ρ ( R ) → L ρ ( R ) ≤ Cε . for all ε ∈ (0 , . As before, R ε : L ρ ε ( R ) → L ρ ( R ) is the unitary transformation R ε ( f )( y ) = ε / f ( εy ). B Appendix: Spectral decomposition of A A is self-adjoint, it has a spectral decomposition and we shall now characterisethe space H + and its dual H − in terms of a realisation of this spectral decomposition. For each θ ,the self-adjoint operator A θ has compact resolvent and for each of its eigenvalues λ n ( θ ), n ∈ N , wedenote by ψ n ( θ ; . ) the corresponding L ρ ( Y )-normalised eigenfunction. Then the mapping Ψ givenby Ψ f ( θ ; · ) = { c n ( θ ) } n ∈ N , c n ( θ ) := Z Y ρ ( y ) f ( θ, y ) ψ n ( θ ; y ) d y, unitarily maps L (cid:0) , π ; L ρ ( Y ) (cid:1) to h := L (0 , π ; ℓ ) so thatΨ (cid:0) U A f (cid:1) ( θ, n ) = λ n ( θ )Ψ (cid:0) U f (cid:1) ( θ, n ) , where for u ∈ h , we denote by u ( θ, n ) is the n -th element of the sequence u ( θ ). It is easy to verifythat Ψ ◦ U unitarily maps H + to h + := (cid:8) u ( θ, n ) ∈ h : (cid:0) λ n ( θ ) + 1 (cid:1) / u ( θ, n ) ∈ h (cid:9) . By standard duality arguments, see for example [16, Chapter 1, Section 6.2], we show that Ψ ◦ U unitarily maps H − , the dual space of bounded linear functionals on H + , to ( cf. (5.54)) h − := (cid:8) f : (0 , π ) → ℓ measurable : (cid:0) λ n ( θ ) + 1 (cid:1) − / f ( θ, n ) ∈ h (cid:9) , in the sense that F ∈ H − if and only if there exists f ∈ h − such that ( cf. (5.55)) H − h F, v i H + = X n ∈ N Z π f ( θ, n ) (cid:0) Ψ U (cid:1) v ( θ, n ) d θ ∀ v ∈ H + , and we have || F || H − = vuutX n ∈ N Z π (cid:12)(cid:12) f ( θ, n ) (cid:12)(cid:12) λ n ( θ ) + 1 d θ. Acknowledgments
KC and SC are grateful for the financial support of the Engineering and Physical Sciences ResearchCouncil: Grant EP/L018802/2 “Mathematical foundations of metamaterials: homogenisation, dis-sipation and operator theory” for KC, and Grant EP/M017281/1 “Operator asymptotics, a newapproach to length-scale interactions in metamaterials” for SC.23 eferences [1] Agmon, Sh.
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