Concentration and non-concentration of eigenfunctions of second-order elliptic operators in layered media

This work is concerned with operators of the type A = --c$\Delta$ acting in domains $\Omega$ := $\Omega$ ' x (0, H) $\subseteq$ R^d x R ^+. The diffusion coefficient c>0 depends on one coordinate y $\in$ (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of ''layered media'', appearing in acoustics, elasticity, optical fibers... Dirichlet boundary conditions are assumed. In general, for each $\epsilon$>0, the set of eigenfunctions is divided into a disjoint union of three subsets : Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as $\epsilon$ $\rightarrow$ 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of ''layered media'' enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y) $\in$ C 2 but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in Fng) are oscillatory. However, this fact by itself does not exclude the possibility of ''flattening out'' of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a ''minimal amplitude hypothesis''. However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem.


INTRODUCTION
Let Ω ⊆ R d , d = 1, 2, . . .be an open bounded smooth domain.In particular, the eigenfunctions of −∆ in Ω form a complete basis in L 2 (Ω ).Our domain of interest is Observe that our regularity assumption on Ω can be considerably relaxed, but this is not the main thrust of the present paper.
In this paper one type of self-adjoint second-order elliptic operators is considered (details are given in Section 2 below) Our study deals with layered media, namely, the diffusion coefficient c depends only on the single spatial coordinate y ∈ (0, H), so that c(x) = c(x , y) = c(y).We use the terminology of diffusion coefficient for lack of a better choice since it appears in the "diffusive term".Note that in the study of the associated wave equation it has the physical meaning of the variable speed of sound.
The dependence of c on a single coordinate results in studying the spectral properties of A via an infinite set of ordinary differential operators with effective increasing potentials (See Remark 2.1).
We always assume homogeneous Dirichlet boundary conditions.Generally speaking, the family of eigenfunctions is split into two categories: those sets of eigenfunctions (or sequences with increasing eigenvalues) involving concentration of mass in proper subdomains of Ω, and those for which such concentration does not occur.
These two categories have been studied by physicists since a long time, investigating diverse phenomena ranging from acoustics to elasticity and to optical fibers 1 .In general, the terminology used in the physical literature has referred to guided or non-guided waves, corresponding, respectively, to concentrating or nonconcentrating modes.We shall use these terms interchangeably, as is appropriate in a particular context.
The reader is referred to [23] for a survey of the geometrical structure of the eigenfunctions of the Laplacian, with very extensive bibliography.
As far back as 1930, Epstein [20] established (in unbounded domains) the existence of acoustic guided waves that are generalized eigenfunctions, i.e. not belonging to the domain of the operator, and are evanescent outside a "guiding channel".The underlying speeds were analytic functions depending on a single vertical coordinate.See [33] for a more general study of Epstein's profiles.An extensive study of guided waves in the acoustic case can be found in [36] and its bibliography.
We mention briefly some other physical instances where guided waves play a significant role.
• The step-index fiber [12].It is a basic model of a cylindrical fiber consisting of a core and external shell ("cladding") carrying two speeds with that of the core smaller than that of the shell.It is a good example of the concentration of the energy in the core.This concentration is increasing when the radius of the core is diminishing.• In optoelectronics much attention is focused on the phenomenon of guided waves, governed by the Maxwell system [12,29].In fact, in the "T E & T M framework the second-order equation for the amplitudes of eigenfunctions [6,Equation (13)] is equivalent to our equation for the amplitude (see below Equation (2.4)).
• The system of linear elasticity in the half-space Ω = R n × (0, +∞) subject to free surface condition.
It gives rise to the Rayleigh surface wave, that is particularly destructive in the case of an earthquake, see [16,34,35].Related phenomena where studied by physicists such as Lamb, Love and Stoneley.Refer also to [15] and references therein.The terms concentration and non-concentration do not always carry the same meaning when used by various authors.The following definition clarifies their meanings in this paper.
For an open set ω ⊆ Ω and v ∈ L 2 (Ω), define is a sequence of normalized eigenfunctions associated with an increasing sequence of eigenvalues and then we say that {v j } ∞ j=1 concentrates in Ω \ ω.On the other hand, if then the sequence is non-concentrating.
Remark 1.2.Later on we shall extend these notions also to sets of eigenfunctions that are not necessarily arranged as such sequences.Note that we study concentration and non-concentration for infinite subsets of eigenfunctions and not necessarily for the whole set of eigenfunctions.
In general, the occurrence of concentration phenomena for second-order operators of the types (1.2) depends on two features: • The shape of the boundary ∂Ω.
• The geometric properties of the diffusion coefficient c(x).
The literature concerning the concentration/non-concentration phenomena as related to the shape of Ω is very extensive.A well-known aspect is the connection of "quantum ergodicity" to "classically chaotic systems" [9,10,11,24,31] and references therein.The paper [32] deals with spherical and elliptical domains.
In contrast, in this paper we are interested in the effects of the layered medium.Thus it is more closely related to the study of operators of the type L = −∇ • (c∇) + V on a finite domain, where the potential V (x) ≥ 0 is positive on a subset of positive measure.Typically, eigenfunctions associated with eigenvalues below ess sup V (x) are concentrating.In [3] the authors replace V by an effective potential u(x) satisfying Lu = 1.They show concentration and exponential decay of eigenfunctions as derived from the geometry of u.Our operator A (1.2) does not involve a potential but the concentration of suitable sequences of eigenfunctions results from the geometry of the diffusion coefficient.As we shall see in Theorem 2.4 below there is a strong underlying geometric aspect; the concentration expresses the fact that the masses of eigenfunctions "flow" (as the eigenvalues increase) into the "wells" (or "valleys").
Turning to the non-concentration case, we observe that the existing literature is less extensive, perhaps due to the fact that it is not directly related to physical or industrial applications.Nevertheless we shall see that it leads to some interesting mathematical questions concerning the structure and asymptotics of eigenfunctions (typically associated with large eigenvalues).Recent publications in this direction are [25] dealing with non-concentration in partially rectangular billiards and [13] concerning piecewise smooth planar domains.A non-concentration result in a stricter sense is that "almost all eigenfunctions of a rational polygon are uniformly distributed" [30].Estimates for nodal sets such as [18] were extended in [26,27] motivated by questions from control theory and [28] that deals with non-concentration in the Sturm-Liouville theory.Note that in the 1-D case issues of non-concentration are closely related to details of oscillatory solutions in the Sturm-Liouville theory.We shall come back to it later in this introduction.
This paper deals with both concentration and non-concentration phenomena for eigenfunctions of layered operators.As already pointed out the latter is less studied in the literature, especially when the diffusion coefficient c(y) is not regular (even discontinuous).As a result, the non-concentration case plays a greater role in this paper.For such eigenfunctions we extend the scope of the study; not only facts pertaining to nonconcentration but a more detailed study of the structure of the solutions in terms of the oscillatory character, amplitudes and their ratios and asymptotic behavior.In contrast to the concentrating case, we shall see that the essential features of the non-concentrating solutions depend primarily on the maximum and minimum of c(x) = c(x , y) = c(y) and, going deeper into the structures, on the total variation of c(y).Our main tool will be the minimal amplitude hypothesis (see Definition 2.6), applied to families of diffusion coefficients.
The paper is organized as follows.
In Section 2 we introduce all relevant notations and details concerning the functional setting.In our case, the eigenvalues are classified by a double-index enumeration, with a conic sector (in index space (µ 2 k , λ), see Figure 3) distinguishing eigenvalues (see (2.9)) associated with concentrating eigenfunctions (F G ) from those (see (2.12)) associated with non-concentrating eigenfunctions (F N G ).This curve serves as the analog to the maximal value of a perturbation potential that separates concentrating from non-concentrating eigenfunctions in the potential perturbation framework.
• Our main result for the concentrating case (F G ) is stated in Theorem 2.4.In particular, it yields exponential decay of the eigenfunctions outside the concentration layers.• In order to deal with non-concentration of certain families of eigenfunctions (F N G ) we introduce the aforementioned minimal amplitude hypothesis.This hypothesis is a geometric assumption on the asymptotic behavior of the amplitudes in the (u, u ) phase plane.The non-concentration of sets of oscillatory solutions follows directly from the geometric assumption (Theorem 2.8).
Section 3 deals with guided waves for A = −c∆.The main result Theorem 2.4 is proved and, on the way, we prove the existence of sequences of eigenvalues satisfying the hypotheses of this theorem (see condition (2.7)).The exponential decay of eigenfunctions is derived from sharp estimates of the Green function.
---------December 13, 2022 In Section 4 we turn to the case of non-concentrating eigenfunctions (non-guided waves in the physical literature) for A = −c∆.The set of corresponding eigenvalues is A c ε (see Definition 2.5) that are located in the aforementioned upper conic sector in the index grid.
The first approach that comes to mind is to transform the problem to a canonical form.In other words, to use coordinate transformations so that the diffusion coefficient becomes a "manageable" perturbation of a constant one.In fact, this is done in Subsection 4.1 under the assumption that c(y) ∈ C 2 ([0, H]).In this case the classical Liouville transformation can be invoked, leading to a detailed asymptotic (almost sinusoidal) behavior of the non-concentrating eigenfunctions.
Once the diffusion coefficient c(y) is less regular, establishing non-concentration becomes considerably more delicate since the classical asymptotic methods are not applicable.Thus, in the rest of Section 4 we focus on proving the minimal amplitude hypothesis that implies Theorem 2.8.Furthermore, the hypothesis is established simultaneously for a full family K of coefficients (see (2.11)).It underlines the fact that only the extremal values of c(y) come into play for Lipschitz continuous or monotone diffusion coefficients.We exploit different methods in handling various classes of functions c(y), such as Lipschitz functions in Subsubsection 4.4.1 or monotone functions in Subsubsection 4.4.2.In each case, additional properties of the solutions are obtained, such as given in Corollary 4.10 for the case of monotone coefficients.The ultimate case where we were able to establish the minimal amplitude hypothesis is for c(y) being of bounded total variation.As a result non-concentration is shown to hold simultaneously for the full family of diffusion coefficients of total variation T V (c) below a fixed V.More specifically we get Consider (for every c(y) ∈ K V ) the subset of eigenvalues A c ε (see (2.12)) and the associated eigenfunctions Then there exists f ω > 0 such that uniformly for all c(y) ∈ K V and all eigenvalues in A c ε .This theorem will be proved as part of the more detailed Theorem 4.13.
Remark 1.3.The uniformity statement in K V is relevant for physical applications, where the coefficient c(y) is only approximately known.
Remark that the case of a continuous c(y), but not of bounded variation, remains an open problem, whence the following question arises naturally: What degree of regularity of c(y) could serve as necessary and sufficient in order to satisfy the minimal amplitude hypothesis (Definition 2.6)?
As already mentioned, the model of piecewise constant coefficients is prevalent in the physical and engineering literature.We have therefore chosen to include Section 5 where we treat in a self-contained way the case of a piecewise constant diffusion coefficient c(y) in both guided and non-guided cases.In this treatment we implement more explicitly some tools that appear frequently in the physical literature, such as detailed expressions for the solutions in layers and their transmission relations across layers.In fact some estimates obtained here are sharper than those derived in Sections 3 and 4.
Appendix A is added for auxiliary results.
In a subsequent paper we shall deal with the concentration and non-concentration issues for operators in divergence form.

SETUP AND MAIN RESULTS
Recall ( (1.1)) that Ω := Ω × (0, H).The coordinates in Ω are designated as x = (x , y) ∈ Ω × (0, H).We introduce a diffusion coefficient (x , y) → c(x , y) such that c(x , y) = c(y) for all x ∈ Ω and of which we shall assume at least the following We focus on the operator A = −c ∆.For the Laplacian −∆ x acting in L 2 (Ω ) with domain H 2 (Ω ) ∩ H 1 0 (Ω ), we denote by (µ 2 k , φ k ) k≥1 the sequence of pairs (nondecreasing sequence of eigenvalues counting multiplicity, normalized eigenfunctions).As the coefficient function c(x , y) = c(y) depends only on the last coordinate y, a separation of coordinates is natural.Using spectral decomposition in the x −coordinate the operator A := −c ∆ is unitarily equivalent to a direct sum of reduced operators in the form The eigenvalues of A are ordered by a two-index system, namely σ(A) = {β k, , k, ≥ 1} where Λ k = {β k,1 , β k,2 , . ..} is the increasing sequence of the eigenvalues of A k .In others words, for each eigenvalue λ of A there exists at least one k ∈ N * such that λ is a simple eigenvalue of A k , whence there exists at least a pair (k, ) ∈ N * × N * such that λ = β k, (there is a one-to-one relationship between the pairs (k, ) and (λ, k)).
Note that in general if λ is not a simple eigenvalue, there is a finite number of pairs (k, ) such that λ = β k, .
We construct an orthonormal basis of eigenfunctions B = {v k, } k≥1, ≥1 associated with the eigenvalues β k, .They are given by v k, (x , y) = φ k (x )u k, (y) where u k, (y) satisfies (2.4) c(y)u k, + (β k, − c(y)µ 2 k )u k, = 0, u k, (0) = u k, (H) = 0. Remark 2.1.As is typical in "separation of variables" situations, the study of the spectral properties of the partial differential operator A is carried out by controlling the behavior of the infinite set of ordinary differential operators of the type (2.4).
Henceforth we use the notation u λ,k instead of u k, .We often write v λ instead of v k, . (2.5) In this paper we are primarily interested in the phenomena of concentration or non-concentration of the mass of eigenfunctions.
• ON THE CONCENTRATION Let ω = Ω α,β be a layer of Ω.We say that ω is a well for the profile c(y) if there exists c 1 > 0 such that (See Figures 1 and 2).
In the concentration case we have the following theorem, which yields exponential decay outside a well.The proof is given in Section 3. Observe that the only hypothesis imposed on c(y) is (2.1).
The exponential decay in estimate (2.8) can be compared to the results of [3].In our proof the 1-D dependence of c(y) enables us to use sharp estimates of Green's kernel.On the other hand in [3] the authors deal with a positive potential perturbation, that leads to a construction of an "effective potential".In terms of this potential the exponential decay is expressed by an "Agmon-type" [1] metric.In our case, from (2.4) we can view the term c(y)µ 2 k as the equivalent of a potential (but unbounded as k → ∞).• ON THE NON-CONCENTRATION.The second type of results concerns the sets (indexed by ε > 0) of non-guided normalized eigenfunctions (the set F N G of the Introduction).They are associated with eigenvalues k , c M := ess sup y c(y).This set is characterized by the fact that there is a positive lower bound for the masses in any layer Ω a,b , uniformly for all its elements.
Recall that λ can correspond to several pairs (k, ) and only some of them satisfy the above inequality.
The geometrical interpretation of non-concentration is clear in the one-dimensional case Ω = (0, L) and λ = β k, > c M µ 2 k : at each interface the angle between the wave and the normal is less than the critical angle stipulated by geometric optics.So, the eigenfunction can travel across each layer without big loss.
---------December 13, 2022 In physical applications it is conceivable that the diffusion coefficient c(y) is known only approximately.It is therefore interesting to extend our study to deal with sets of such coefficients.Let 0 < c m < c M be fixed.We assume that every coefficient c(y) satisfies condition (H) (see (2.1)) and denote by the family of all such coefficients.In various cases, we shall impose further assumptions on the elements of K .
We introduce the set of eigenvalues as above, whose associated eigenfunctions will be shown to be (perhaps under additional assumptions) non-concentrating.Definition 2.5.Fix ε > 0. For any fixed µ k , let 0,k be the first satisfying k .We designate (see Figure 3) Next we define the minimal amplitude of the family of the associated solutions as follows. (2.13) In the subsequent discussion the parameter ε > 0 is fixed and for simplicity of notation we omit the indication of the dependence of r c on it.Definition 2.6.Let c(y) ∈ K .We say that c(y) satisfies the minimal amplitude hypothesis with respect to A c ε if Remark 2.7.Note that this hypothesis has a very clear geometric interpretation by means of the Prüfer substitution [7].
Observe that while the minimal amplitude deals with the sum of squares u λ,k (y) 2 + u λ,k (y) 2 , the non concentration involves only the integral of u λ,k (y) 2 over various intervals.The following Theorem 2.8 connects these topics, showing that the minimal amplitude hypothesis implies non-concentration.Here we state it using the physical model with the spectral parameter λ.It is proved in a somewhat more detailed form (using the reduced eigenfunctions u λ,k ) as Theorem 4.5 in Subsection 4.2.
Theorem 2.8 (Non-concentration in any layer).Let c(y) ∈ K be a diffusion coefficient satisfying the minimal amplitude hypothesis.For any Then there exists a constant C ω > 0 such that, Remark 2.9.We shall see that in various cases we can find subsets K 1 ⊆ K such that the inequality (2.15) holds uniformly with respect to c ∈ K 1 .
• In Section 4 we show that any eigenfunction associated with eigenvalues in A c ε behaves in an oscillatory fashion.This is a straightforward consequence of the comparison principle.However, it does not exclude the possibility that some of the sections of the oscillatory solution may "flatten out", namely their amplitudes shrink as λ → ∞.The condition (2.13) ensures that such phenomena do not happen, as is stated in Theorem 2.8.
• In Subsection 4.2 we discuss the meaning of the minimal amplitude hypothesis.If the function c(y) is of bounded total variation we prove (Theorem 4.13) that it satisfies the hypothesis with respect to A c ε .This covers the cases of functions in C 1 ([0, H]) as well as functions in W 1,1 ([0, H]), piecewise constant functions ...

GUIDED WAVES (SEE THEOREM 2.4)
We refer to the geometric setup in Definition 2.3 above.The simplified case with α = 0, β = h 0 is common in the physical literature dealing with band structure.It was our starting point at the early stage of this work [5].In this section we always assume (2.1).

PROOF OF THEOREM 2.4.
Using the notation in Definition 2.3 and (2.5) we are interested in the behavior of u λ,k (solution to (2.4)) as λ → ∞.Note that for all (λ, k), λ = β k, , the function w λ,k (y) := u λ,k (y) 2 is a solution to (where we use w for simplicity) Some properties of the solutions of (3.1) We now derive upper and lower bounds for solutions of (3.1).
Claim 3.1 (Upper pointwise bounds for the geometric situation as in Definition 2.3).Let w be a solution to Proof.For simplicity of the presentation we take α = 0. We use the Green function G of the Dirichlet operator w − ξ 2 w and prove exponential decay outside the well if λ < c 1 µ 2 k , depending on the distance of y to the well.The Green function is given by (3.4) ∀y, y ∈ (0, H), G(y, y ; ξ) In view of (2.6) one has c(y) ≥ c 1 outside (0, β) and, as f is nonpositive on (0, H) \ (0, β) and G ≤ 0, this implies The estimate (3.5) becomes w(y) ≤ −2λ Proof.Again for simplicity we take α = 0. Taking into account that w(H) = w (H) = 0, if a > β, we integrate twice (3.1) from H. As (u k,λ ) 2 ≥ 0 and f ≤ 0 on (β, H), we obtain with Remark 3.3.In the estimates above we have used the explicit form of the Green kernel.As an alternative we could use general trace estimates that are applicable also for divergence-type operators −∇ • (c∇), where an explicit kernel is not available.However, this method yields only a polynomial rate of decay ).This approach will be used in a subsequent paper.
Observe that if the profile c(y) has two wells, with the same "depth" c m (see Definition 2.3), then the method of proof of Theorem 2.4 fails.However, by enlarging (α, β) so that it contains the two wells, we can repeat the proof to get concentration in this extended band.Remark 3.4 (Estimating in terms of a subdomain of the well).Note that in the right-hand side of the estimate (2.8) the mass in the well is Suppose that there exists an open domain ω ⊆ Ω and a subsequence (retaining the same index)

EXISTENCE OF EIGENVALUES COMPATIBLE WITH ASSUMPTION (2.7).
The previous results rely on the existence of infinitely many eigenvalues satisfying Assumption (2.7).This fact is established in the following theorem.
Theorem 3.5.The number of eigenvalues satisfying Assumption (2.7) goes to infinity with k.
Second, for each k ∈ N * the smallest eigenvalue β k,1 of A k is given by inf . Third, we know that inf dy 2 defined on U with Dirichlet boundary conditions.So, we can write Then the sequence (β k,1 ) k satisfies Assumption (2.7) for k > K.This proof exhibits only a sequence but we can build other sequences satisfying this assumption.We skip a detailed discussion of this fact for the sake of brevity.

NON-GUIDED WAVES
An (infinite) set of non-guided normalized eigenfunctions is characterized by the fact that in each layer Ω a,b there is a uniform positive lower bound for the masses in the layer, valid for all elements of the set.
As observed in the introduction, for each eigenvalue λ of A, there exists at least one pair (k, ) so that λ = β k, is the -th eigenvalue of A k .Let λ = β k, > 0 be an eigenvalue of −c(y)∆ in L 2 (Ω, c(y) −1 dx dy) and u(y; λ, k) := u λ,k (y) the normalized associated (reduced) eigenfunction (as in (2.4) and (2.5)).The function u(y; λ, k) satisfies We shall deal in this section with eigenvalues λ such that (see (2.12)) In particular, for such values we have p(y; λ, k) > 0.
A desirable way to treat this equation is by transforming the equation into a canonical equation of the type with some new variable ξ and new unknown η.
The aim of the subsequent subsections is to claim that the set of eigenfunctions associated with eigenvalues satisfying (µ k , λ) ∈ A c ε , for any diffusion coefficient c(y) ∈ K (see (2.11)) consists of non-guided eigenfunctions when a particular sufficient condition is satisfied, with c(y) less regular.
Consider a pair (µ k , λ) ∈ A c ε .Let u(y; λ, k) be a normalized solution to (4.1), associated with (µ k , λ).In view of (2.12) The following claim extends (4.3) and will be useful in the sequel.It says that the distance between two consecutive zeros of an (oscillatory) eigenfunction can be made arbitrarily small, if we drop a finite number of eigenfunctions associated with "low" eigenvalues.The threshold λ α,ε applies uniformly to all coefficients c(y) ∈ K .Claim 4.1.Let c(y) ∈ K .For each α > 0, ε > 0 there exists λ α,ε such that λ > λ α,ε implies In particular, λ α,ε can be chosen uniformly for all c(y) ∈ K .For each c(y) ∈ K there are at most finitely many eigenvalues λ < λ α,ε .
Proof.Recall that we are assuming (µ k , λ) ∈ A c ε so that by (4.2) p(y; λ, k) ≥ ε c M .Thus by the comparison principle it suffices to compare (4.1) with the constant coefficient equation v + γ 2 v = 0 for any large γ.Pick Clearly for any λ > λ 0,ε and µ k ≤ µ 0,ε we have µ 2 k p(y; λ, k) > γ 2 .Note that there are at most finitely many pairs (µ k , λ) In what follows we assume that c(y) ∈ K and consider spectral values (µ k , λ) ∈ A c ε .The following claim, an immediate consequence of (4.2), will be useful in the sequel, when estimating masses of eigenfunctions in intervals.
Claim 4.2.Let u(y; λ, k) be a solution to (4.1), where (µ k , λ) ∈ A c ε .Then it is strictly convex (or concave) in every interval In particular, without further assumptions, the solutions are oscillatory and are convex (or concave) between consecutive zeros.However, in various sub-intervals their amplitudes might decay to zero, hence concentrating in the complementary domain.It is precisely this behavior that we seek to exclude.
We start off with the classical case of a C 2 coefficient c(y).In this case, a full asymptotic characterization of the eigenfunctions is possible.
Observe that under our assumptions the family The uniform boundedness of the family B implies that the Volterra integral equation (4.8) is solvable for any sufficiently large µ k and furthermore, for any small δ > 0, there exists µ 0 > 0 so that (4.9) We now make the following observations.
Remark 4.4.Note that the hypotheses of Theorem 4.3 entail not only the conclusion that the eigenfunctions do not concentrate in sub-domains of (0, H) but also their asymptotic (sinusoidal) form, as in (4.9).

BEYOND THE REGULAR CASE (SEE THEOREM 2.8).
The implications of the assumption that c(y) is subject only to the minimal amplitude hypothesis ( Definition 2.6) will now be studied.No regularity is required of c(y), and only condition (H) (see (2.1)) is imposed.
We have already seen that the lack of regularity does not affect the oscillatory character of the solutions.The remaining issue is to see that the masses of the oscillatory solutions {u(y; λ, k), (µ k , λ) ∈ A c ε } in any interval remain uniformly bounded away from zero.This is addressed in the following theorem which is a somewhat more detailed form of Theorem 2.8.Its proof is straightforward, reducing the non-concentration issue to a study of the minimal amplitude hypothesis for various functional classes.This estimate is equivalent to Ω a,b v λ (x) 2 c(x) −1 dx ≥ d, where v λ is the eigenfunction of −c∆ associated with λ.

MORE ON THE MINIMAL AMPLITUDE HYPOTHESIS.
Recall that the minimal amplitude was defined by (2.13): We now consider this quantity in more detail.
Proposition 4.6.Let u(y; λ, k) a normalized solution to (4.1) where c(y) ∈ K and Then there exists a positive λ0 such that for any λ > λ0 , Proof.Suppose that u(y; λ, k) is convex in the interval (z i , z i+1 ) (see Claim 4.2).In light of Equation (4.1) the function u(y; λ, k) satisfies In the interval we have u(y; λ, k) < 0 and by convexity As in the proof of Claim 4.1, we can now find λ0 so that µ2 k p(y; λ, k) > 1 for λ > λ0 .Inserting this in (4.21) we obtain In this subsection we present several subsets of the set K of diffusion coefficients c(y) for which the minimal amplitude hypothesis can be verified.In fact, our ultimate subset is that of functions of bounded total variation (see Subsection 4.5), that contains all the subsets considered here.To justify our special treatment of the more restricted subsets, we call attention to the following points.
• As we narrow down the admissible coefficients c(y) (as we did above for c(y) ∈ C 2 ) we can extract more information on the general structure of the corresponding non-guided eigenfunctions.• The methods of proof in the different cases are quite different from each other.Given the important role of the minimal amplitude in these investigations and the fact that the most general case (namely all c(y) ∈ K ) is still open, it seems to us worthwhile to expound the various methods.• The special case of piecewise constant coefficients is in the focus of much of the physical literature, and some of the estimates obtained in this context will prove to be crucial in establishing the more general case.

c(y) LIPSCHITZ
The first subset to be considered in the following proposition is that of Lipschitz functions.
Then it satisfies the minimal amplitude hypothesis with respect to A c ε .Proof.Let u(y; λ, k) be a normalized solution to (4.1).We just need to prove the estimate (2.14).Equation (4.1) can be rewritten as Observe that in light of (4.2) We now replace u, u by u 1 , u 2 as follows (suggested in the recent paper [2]).
and note that the vector function Since u2 1 + u 2 2 > 0 in the interval [0, H] we conclude that there exists a constant R > 0, so that for all (µ k , λ) ∈ A c ε and for any Furthermore, since u(y; λ, k) is normalized H 0 u(y; λ, k) 2 c(y) −1 dy = 1, it follows that there exists a constant η > 0, so that for all (µ k , λ)

c(y) MONOTONE NONDECREASING
In the following proposition we relax the regularity of the coefficient c(y).In fact it is no more required to be continuous but on the other hand a monotonicity assumption is imposed.Proposition 4.9.Let c(y) ∈ K and assume that c(y) is nondecreasing.Then it satisfies the minimal amplitude hypothesis with respect to A c ε .Proof.Let u(y; λ, k) be a normalized solution to (4.1) where (µ k , λ) ∈ A c ε .Define r c,µ k ,λ as in (4.18) and use the transformation (as in the proof of Proposition 4.8) We first show that even though α(y; λ, k) is not necessarily continuous, Equations (4.26)-(4.27)can be extended (in distribution sense) so that (4.31) Indeed, we first have H] be a uniformly bounded sequence converging to α(y; λ, k) a.e.(hence in distribution sense) 2 .We may also assume that it is uniformly bounded away from zero.Under these conditions the sequence u (y;λ,k)  As above, let be the set of zeros of u(y; λ, k).Recall that the set z i+ 1 2 is defined as in (4.13).
From Proposition 4.6 and ( * ) we deduce that there is λ0 such that for any λ > λ0 , (4.32) u(z i+ 1 2 ; y; λ, k) 2 ≤ u(z i+ 3 2 ; y; λ, k) 2 , 0 ≤ i < s − 1 and (4.33) r 2 c,µ k ,λ = u(z 1 2 ; y; λ, k) 2 .The proof will be complete if we prove Indeed, there are only finitely many eigenfunctions with (µ k , λ) ∈ A c ε and λ ≤ λ0 , hence Note that by excluding at most a finite number of eigenvalues we shall be able to obtain a more explicit lower bound for the rc in (4.34).This will be evident in Corollary 4.10 below.
It remains to prove (4.34).This is done in two steps.
First, we fix ε > 0, 0 < c m < c M , and define K , A c ε as in (2.11), (2.12), respectively.Let K 1 ⊆ K be the set of all nondecreasing diffusion coefficients.
Observe that the threshold value λ0 depends only on ε, c m , c M .Also the constant c appearing in (4.38) depends solely on these parameters.Corollary 4.10.
• The minimal amplitude for all solutions with λ > λ0 is strictly positive: • The amplitudes of any solution u(y; λ, k) between zeros are growing as y moves from 0 to H.However, the ratios of the amplitudes remain universally (for all c(y) ∈ K 1 ) bounded for λ > λ0 .

c(y) PIECEWISE CONSTANT
We turn next to the case that c(y) is a piecewise constant function.In Theorem 4.13 below we discuss our most general case, namely c(y) of bounded variation.To this end, a detailed treatment of the piecewise constant case is needed.
We show that c(y) satisfies the minimal amplitude hypothesis, where the relevant constants depend only on its total variation.Notational comment: In order to keep the notational uniformity with the other sections, we retain the notation c m , c M for the minimal and maximal values, respectively, of c(y).Of course they coincide with some c j s but the distinction in various estimates (such as (4.52)) will be completely clear.
Recall that u(y; λ, k) satisfies Equation (4.1) with p(y; λ, k Proposition 4.11.Assume that c(y) is piecewise constant as above, and let be the total variation of c(y).Let u(y; λ, k) be a normalized solution to (4.1) where (µ k , λ) ∈ A c ε .

Then
(1) c(y) satisfies the minimal amplitude hypothesis with respect to Note in particular that d and λ 0 do not depend on the size N of the partition.
Proof.In light of Theorem 4.5 we need first to prove the validity of the minimal amplitude hypothesis.Consider an interval I j = (h j , h j+1 ).
The estimates (4.46) and (4.47) imply that the minimal amplitude hypothesis is satisfied The non-concentration estimate (4.44) is now a consequence of the general Theorem 4.5.
In analogy with the case of the subset of all nondecreasing coefficients (Corollary 4.10) we deduce a similar result for all piecewise constant coefficients having a uniform bound of their total variations.Corollary 4.12.Let K P CV ⊆ K be the set of all piecewise constant diffusion coefficients, with total variation less than V. Then Our ultimate result concerns the case that Recall that K was defined in (2.11).
We establish non concentration for spectral pairs (µ k , λ) ∈ A c ε (see (2.12)).As in the cases studied above, the proof relies on the validity of the minimal amplitude hypothesis, via the fundamental Theorem 4.5.
Theorem 4.13.Let c(y) be of bounded variation.Then it satisfies the minimal amplitude hypothesis with respect to A c ε , uniformly for all {c(y) ∈ K , T V (c) ≤ V } .More precisely, as in (2.13), (4.55) Furthermore, for any (a, b) ⊆ (0, H) there exists a constant f a,b > 0 such that, for every c(y) ∈ K V , and for every normalized u(y; λ, k) associated with Then there exists f ω > 0 such the eigenfunction v λ (x , y) = u(y; λ, k)φ k (x ) satisfies uniformly for all c(y) ∈ K V and all eigenvalues in A c ε .
The proof consists of approximating c(y) by a sequence of piecewise constant functions and using the results of Proposition 4.11 and Corollary 4.12.The approximation procedure is based on the following result [8, pp.12-13].
Claim 4.14.Suppose that c(y) ∈ K and is of total variation V > 0. Then there exists a sequence of piecewise constant functions c with associated operators For the Laplacian −∆ x acting in L 2 (Ω ) with domain H 2 (Ω ) ∩ H 1 0 (Ω ), we denote by (µ 2 k , φ k ) k≥1 the sequence of normalized eigenfunctions and their associated eigenvalues, ordered by µ k ≤ µ k+1 .The eigenfunctions of A (resp. We consider eigenfunctions associated to spectral pairs (µ k , λ) ∈ A c ε .The following perturbation lemma is at the basis of the proof of the theorem.We postpone its proof to the end of this section, following the proof of the theorem.Note that in this lemma no assumption is needed concerning the total variations of the involved functions.n) and A be the corresponding operators.Let λ > 0 be an eigenvalue of A, with associated normalized eigenfunction u(y; λ, k)φ k (x ).Then there exist N > 0 and a sequence of eigenvalues Proof of Theorem 4.13.Pick some λ ∞ n=N be a sequence as in Lemma 4.15.Note that the convergence (4.62)(i) implies that, for sufficiently large index n the condition λ (n) > (c M + ε 2 )µ 2 kj holds.In view of the uniform bound (4.60) on total variations we can invoke Corollary 4.12 to get (4.63)(r app c ) 2 := inf where L > 0 depends only on ε, V, c m , c M (see (4.53)).The H 2 convergence (4.62)(ii) entails uniform convergence of both the functions and their derivatives.Hence The estimate (4.55) now follows from the fact that, in view of Corollary 4.12, the estimate (4.63) holds uniformly for all approximating sequences for any solution u(y; λ, k) associated with (µ k , λ) ∈ A c ε .In fact, we get the uniform estimate (4.56) since r c depends only on V. Due to (4.56) this estimate is uniformly valid (with the same d, λ 0 ) for all c(y) ∈ K with T V (c) ≤ V.
However, for every c(y) ∈ K with T V (c) ≤ V there are finitely many eigenfunctions that are excluded, namely, those with λ < λ 0 .Clearly, these eigenfunctions vary with c(y).We now show that they can be included in (4.57).The price to be paid is that the lower bound f depends in a more delicate way on the various parameters (and not only on ε, c M , c m , b − a, r c ).
---------December 13, 2022 To obtain a contradiction let {c n (y), T Assume further that (c M + ε)µ 2 kn < λ n < λ 0 , n = 1, 2, . . .Suppose that for some interval (0, H) and some subsequence (we do not change indices) This is a contradiction to the fact (see (4.56)) Proof of Lemma 4.15.We use the direct sum representation (2.2) both for the operator A and the operators A (n) .Since the eigenfunctions {φ k (x )} do not depend on the index n, the reduced operators A (n) k (see (2.3)) are given by Fix k ∈ N * so that λ is an eigenvalue of A k .The corresponding (reduced) eigenfunction u(y; λ, k) satisfies the equation (see (4.1)) Let B(λ, δ) ⊆ C be the disk of radius δ centered at λ and consider the following linear initial value problem, with a complex parameter z ∈ B(λ, δ), For every y ∈ [0, H] the function w(y; z) is analytic as a function of z [14, Chapter 1, Th.8.4] and this is true in particular for f (z) := w(H; z).Note that z is an eigenvalue of A if and only if f (z) = 0, since if w is an eigenfunction then so is aw, for any a = 0. Clearly f (λ) = 0.This is the only zero of f in B(λ, δ) for sufficiently small δ > 0, since λ is an isolated eigenvalue of A k .
By standard formulas for zeros of analytic functions, since λ is a simple zero, and let z (n) ⊆ B(λ, δ) be a sequence converging to z.The Rellich compactness theorem yields the existence of a subsequence w (nj ) ∞ j=1 and a limit function w(y; z) such that (4.70) lim j→∞ w (nj ) (y; z (nj ) ) = w(y; z), lim j→∞ w (nj ) (y; z (nj ) ) = w (y; z), strongly in L 2 (0, H), and w (nj ) (y; z (nj ) ) converges strongly to w (y; z) due to the equation itself.It follows that w(y; z) satisfies (4.68) with the same initial data, so by uniqueness w(y; z) = w(y; z).In particular, since all converging subsequences have the same limit, (4.70) can be replaced by where the real sequence λ (n) satisfies lim n→∞ λ (n) = λ.
In addition, λ (n) (for sufficiently large n) is an eigenvalue of ) is an associated (not necessarily normalized) eigenfunction.To conclude the proof of the lemma we take .

THE DIFFUSION COEFFICIENT IS PIECEWISE CONSTANT-DETAILED STUDY
There is special physical interest in the case that the diffusion coefficient is piecewise constant.For this reason, we focus here on this case, providing detailed information for both guided and non guided waves.Of course, in this case c(y) is of bounded variation, hence the results of Theorem 2.4 and Theorem 4.13 are applicable.However, we get here more detailed estimates by using more direct methods.
Notational comment: As in Subsection 4.4.3, in order to keep the notational uniformity with the other sections, we retain the notation c m , c M for the minimal and maximal values, respectively, of c(y).Of course they coincide with some c j s but the distinction in various estimates will be completely clear.
This particular case is related to optical fibers for their industrial applications in both acoustics and optics and to printed circuit boards.The simplest example of an optical fiber is the step-index fiber: the fiber has two cylindrical parts sharing the same axis: a core of radius a surrounded by a ring of thickness b, the cladding.A buffer and a jacket protect these two elements by surrounding them.The index of the core is n = n 1 > 0 and that of the cladding are n = n 2 < n 1 .Our coefficient of diffusion c is exactly c = 1 n .According to the choices of the respective constants a, b, n 1 , n 2 and of the used pulse, the fiber is a single-mode fiber or a multiple-mode fiber.The non-specialist reader interested in these modes of data transport could consult many sites 4 .
The cross-section of a fiber is a disk that we match to our open set Ω := (0, L)×(0, H) which are therefore not diffeomorphic.The analogy between the step-index fiber and this subsection is achieved by taking N = 1, c 0 < c 1 in the notations that follow.A pulse being fixed, a link between our work and the properties of the optical fibers is the following one: in the both cases we reduce the question to a one-dimensional problem by separation of variables, replacing our variable y by r, the distance to the center of the disk.For the fiber the problem reduces to a family of Bessel equations according to modes E, H, ... and the chosen simplifications whereas it is (2.2) for us.This correspondence has certainly theoretical limits: for the Dirichlet Laplacian in a disk there are eigenfunctions associated to high eigenvalues that are concentrated close to the boundary of the disk (see [32,Section 7.7] , [23]).As a matter of fact, the frequencies not going to infinity in applications, this influence is reduced.

GUIDED WAVES FOR MONOTONE PIECEWISE CONSTANT c(y).
This self contained subsection is a special case of Section 3 by assuming c is piecewise constant and monotone increasing when 0 < y < H, which is the common structural assumption in physical applications, in particular in studies of optical fibers.With the above-mentioned precautions we are therefore dealing with the analogous case of the graduated-index fibers when the index n is piecewise constant and we prove the existence of these specific modes that are evanescent in the cladding.In our problem we find again the same properties of concentration of energy in the first layers of Ω.This concentration is increasing when their thickness decreases.
We begin by listing the hypotheses in this part (see Figure 4).
Let us fix i ∈ {1, • • • , N } and assume that, for a certain pair (k, ), the eigenvalue λ Then, for an eigenfunction u λ,k associated to β k, , the function w = u 2 λ,k is a solution of (3.1) with f given by (5.1) and g as in (3.1).Then the definitions of G and f (see (5.1)), combined with the monotonicity of c and y < x Proof.Taking into account that w(H) = w (H) = 0, w integrating twice (3.1) and taking into account that (u k,λ ) 2 ≥ 0, we obtain

Proposition 5.1 (Upper pointwise bounds). There exists
Proof.It suffices to apply Propositions 5.1 and 5.2 with C = While the estimate (5.2) is more precise than (2.8), it implies the same type of exponential decay of the eigenfunctions v λ (x) = φ k (x )u λ,k (y) as follows.
and each eigenfunction v λ , one has (see Definition 2.2) The reader may wonder if the concentration takes place only in the layer Ω hi−2,hi−1 , the concentration in the other layers Ω hip,hip+1 , 0 ≤ p ≤ i − 3, becoming negligible when k → ∞ ?Theorem 5.5 is a counter-example with the eigenfunctions v λ = φ k (x )u λ,k (y) where Theorem 5.5.Let c(y) be piecewise constant, taking three increasing values c 0 < c 1 < c 2 and let {v λn } ∞ n=1 be a sequence of eigenfunctions associated with the eigenvalues λ n = β kn,ln satisfying (c Then the L 2 norms of the eigenfunctions v λn concentrate in Ω 0,h0 ∪ Ω h0,h1 (the lower two layers) when n → ∞ as follows.
Proof.The following transmission conditions hold  −h1) .This concludes the proof.
It remains to be proved that the condition in Theorem 5.5 is not void, namely, that for each index i and each k sufficiently large there exists at least one eigenvalue β k, located in (c i−1 µ 2 k , c i µ 2 k ).This is proved in the following theorem subject to an additional hypothesis which restricts the class of operators considered.Note that Theorem 3.5 does not guarantee that eigenvalues β k, are included in the interval (c i−1 µ 2 k , c i µ 2 k ).The additional hypothesis mentioned above is sufficient to obtain this fact.
Theorem 5.6.For ε > 0 sufficiently small, a sufficient condition for the existence of an infinite sequence of eigenvalues Note that the inequality (5.12) requires c 2 1 < c 0 c 2 .For the proof, see Appendix A. Remark 5.7.
(1) In this Subsection, we have considered a monotone increasing function c.So, the concentration takes place in the union of layers such that (x , y); y < inf z { λ µ 2 k + ε < c(z)} for a guided eigenvalue λ = β k, .
(2) Wilcox [36] studied similar stratified media but the operator −c(y)∆ acted in R n+1 or R n+1 + whence the point spectrum was empty.Idem in [17] where eigenvalues could appear by perturbing the coefficient c.The concentration in a layer needed a local minimum of c in this layer.
(3) In the last page of [4], we pointed out that, if ξ 1 → 0, the concentration could take place in the layer Ω 1 but it is not clear that the phenomenon could actually take place.
Define a set of n + N functions {ϕ p } n+N p=1 ⊆ H 1 0 (0, H) by: ϕ p (y) = We conclude that there are at least n eigenvalues less than or equal to c 1 µ 2 k .Applying a similar argument to the full set {ϕ p } n+N p=1 we infer that where in the final estimate we used the assumptions on h, h.
As above, we conclude that, for a fixed k there are at least n + N eigenvalues of A k smaller than or equal to (c 2 − ε)µ 2 k .
Step 2 : Existence of eigenvalues of A k between (c 1 + ε)µ Such an integer exists if It is readily seen that (A.4) holds for sufficiently large µ k if (A.5) The validity of (A.5) with ε = 0 follows exactly from the condition (5.12).By continuity, (A.5) will be satisfied for ε < ε 0 , for sufficiently small ε 0 > 0.

4. 1 .
THE REGULAR CASE: THE LIOUVILLE TRANSFORMATION WITH c(y) ∈ C 2 ([0, H]).This case is of interest, as it yields an almost sinusoidal behavior of the eigenfunctions, not only estimates on the mass in a band.Theorem 4.3.Let Λ > ε and set

4 . 4 . 2 .
H].This estimate, combined with the definition of u 1 , u 2 and (4.25) implies the required estimate (2.14) and concludes the proof of the proposition.Second case.