Localized Modes of the Linear Periodic Schrödinger Operator with a Nonlocal Perturbation
We consider the existence of localized modes corresponding to eigenvalues of the periodic Schrödinger operator with an interface. The interface is modeled by a jump either in the value or the derivative of and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be different. As we show, eigenvalues can only occur in spectral gaps. We pose the eigenvalue problem as a gluing problem for the fundamental solutions (Bloch functions) of the second order ODEs on each side of the interface. The problem is thus reduced to finding matchings of the ratio functions , where are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Prüfer transformation. The limit values of at band edges depend on the ordering of Dirichlet and Neumann eigenvalues at gap edges. We show that the ordering can be determined in the first two gaps via variational analysis for potentials satisfying certain monotonicity conditions. Numerical computations of interface eigenvalues are presented to corroborate the analysis.
Localization for perturbed periodic Schrödinger operators , where is periodic in , is a classical problem traditionally treated by spectral theory. Most commonly it is studied for perturbations that are either compactly supported, see, e.g., Deift & Hempel , Alama et al. , and Borisov & Gadyl’shin  or fast decaying, e.g. , cf. Želudev  and Alama et al. . Both of these scenarios can lead to eigenvalues of and thus to localization. Potentials describing random perturbation also yield eigenvalues due to Anderson localization, studied, for example, by Kirsch et al.  and Veselić . We investigate localization in the one-dimensional case due to the presence of deterministic interfaces which cannot be represented as localized perturbations of . Such an interface arises, for instance, when is periodic on one side of the interface and vanishes on the other side (we assume commensurability of the periods of and to preserve periodicity on each side of the interface). This topic has been previously studied mainly by Korotyaev via spectral theory [13, 14]. We, on the other hand, use the properties of the fundamental solutions of the 1D spectral problems of the periodic operators corresponding to each side of the interface and pose the eigenvalue problem as a -gluing problem for the decaying Floquet-Bloch solutions from either interface side. This approach allows us to provide some concrete conditions on and the perturbation directly (without conditions on the spectrum of ) that ensure eigenvalue existence in the semi-infinite and the first finite gap of the continuous spectrum of . Our approach is also arguably conceptually simpler than that of [13, 14].
Localized waves at interfaces of two periodic (linear) structures have been also demonstrated experimentally in the context of electron waves in crystals by Ohno et al.  and for optical waves in photonic crystals by, e.g., Suntsov et al. .
In detail, within the framework of the eigenvalue problem
we study the following two interface problems. Firstly, an interface made of even periodic potentials
where has period , i.e., for all , and furthermore satisfies for all . Secondly, an interface made of dislocated even periodic potentials
where has period , i.e., for all , and satisfies for all . The dislocation parameters are . Here is the characteristic function. Note that under the periodicity conditions the evenness of and about and within the periodicity cell and , respectively, is equivalent to evenness of and about . Hence, in the following we will simply require that the potentials be periodic and even. Unless otherwise stated, the potentials and are continuous and hence bounded.
One of the simplest examples of the interface (1.2) is the additive interface
generated by merely changing the average value of the potential on one half of the real axis. This example is studied in more detail in Section 3.1.1 since the conditions on eigenvalue existence become rather specific in this particular case.
Equation (1.1) finds applications in many fields of natural science. Perhaps most notably it describes the wave function of an electron in a one dimensional crystal, where waves localized at a crystal interface are typically called Tamm states . The equation also directly applies to the description of light propagating transversally to the direction of periodicity of a non-dispersive, lossless, linear photonic crystal which is homogeneous in the and directions. Suppose the refractive index varies periodically in the direction and its mean has a jump at , such that . We assume the following form of the electric field,
such that the field is polarized in the direction, the waves propagate in the direction and the profile is stationary. Then Maxwell’s equations exactly reduce to
With and setting , we recover (1.1).
Another example of an application of (1.1) is the description of matter waves in one dimensional Bose-Einstein condensates loaded onto an optical lattice, see Choi & Niu . The density of a condensate is described by the wavefunction governed by the Gross-Pitaevskii equation [4, 9, 17]
where, in our setting, is periodic but has a jump at . Here is Planck’s constant, is the boson mass, is the potential induced by the optical lattice and is the scattering length. In the linear regime, , stationary waves obey (after rescaling) equation (1.1).
The rest of the paper is organized as follows. In Section 2 we review the needed facts on spectral properties of the interface-free periodic Schrödinger operators with an even potential including the problem of ordering of spectral band edges according to even/odd symmetry of the Bloch functions. Section 3.1 discusses the interface (1.2) and introduces the main tools of our analysis, namely the -matching condition and the Prüfer transformation. The theory is then applied to the additive interface example (1.4) and numerical computations of point spectrum are performed. In Section 3.2 we analyze the dislocation problem (1.3) for the cases and using the same tools as in Section 3.1 plus differential inequalities and variational methods. Numerical examples are, once again, provided.
2 Spectrum of the Interface-Free Problem
We review, first, some well known results on the spectrum and the eigenfunctions of the interface-free operator , where is continuous and . Good sources on the theory of the periodic Schrödinger operator are Magnus and Winkler , Eastham  and Reed & Simon .
has a purely continuous spectrum (see Theorem XIII.90 in ) consisting of bands so that
where and . When , we say that has the finite gap . Clearly, has also the semi-infinite gap . According to Floquet theory  the spectrum can be easily found via the use of the monodromy matrix of the second order ODE . Figure 2 presents the numerically computed spectrum of the operator with .
The ODE has two linearly independent solutions, so called, Bloch functions. For real they are of the form
where if and if , and are real-valued and periodic in . In fact, are either periodic of anti-periodic. If , the Bloch functions are of the form
where again are real and periodic in .
The evenness of the potential and the fact that only one linearly independent bounded Bloch function (namely ) exists at any imply that this solution must be even or odd and hence it satisfies at the boundary-points and either Dirichlet- or Neumann-boundary conditions. For let denote the -th Dirichlet eigenpair of on satisfying and let be the -th Neumann eigenpair of on such that . The following lemma may be well known, cf. , Theorem 1.3.4.
For the first gap edge we have . If and if then , . Moreover, the following properties of the eigenfunctions are known (note that the even/odd-property applies with respect to reflection about ):
Remark. Note that can never be a Dirichlet or Neumann eigenvalue since any corresponding eigenfunction could be extended to a bounded solution of on by reflection and periodic extension. Such nontrivial solutions cannot exist for by (2.1).
As we show in Sections 3.1 and 3.2, ordering between the Dirichlet and Neumann eigenvalues and plays an important role for existence of interface eigenvalues. It is, however, known that all orderings are in general possible, i.e., for any given ordering of the Dirichlet and Neumann eigenvalues (respecting the condition ) a corresponding even potential exists, see Theorem 3 in Garnett & Trubowitz . Nevertheless, the following lemma provides an ordering of low eigenvalues under some monotonicity assumptions on the potential .
If is strictly increasing on , then . The Neumann eigenfunction corresponding to is strictly monotone on and odd with respect to .
If is strictly decreasing on , then . The Dirichlet eigenfunction corresponding to is strictly monotone on and even with respect to .
The proof is based on the following result.
Consider a potential (not necessarily periodic, even or continuous) and let be the first eigenvalue of on with the boundary condition , whereas denotes the first eigenvalue of the same differential operator but with boundary conditions . Then
Moreover, if is strictly increasing on then and any eigenfunction for with is strictly decreasing on . If is strictly decreasing on then and any eigenfunction for with is strictly increasing on .
The proof is inspired by a similar result in Bandle et al. . Note first that the set, on which the minimization is performed, is weakly closed in due to the compact embedding . Hence a minimizer of the right-hand side of (2.3) exists. We denote it by . Let us also denote the value of the minimum by . The proof is now divided into five steps:
Step 1: has exactly one zero on . Since possesses at least one zero , we have . Clearly is then the minimizer of
and therefore satisfies the Euler-Lagrange equation
with boundary condition
where in case one of the two Neumann conditions is dropped. Note that
Now assume for contradiction that has a second zero . Then (2.6) holds also for all and since , we find that (2.6) holds for all , i.e., is a Neumann-eigenfunction. The same applies for , which is also a minimizer of (2.3). But then must be the first Neumann-eigenfunction of on and it therefore has no zero on . This contradiction shows that has exactly one zero in .
Step 2: is strictly less than the second Neumann-eigenvalue on . Since the second Neumann eigenfunction has one zero in , we find . Suppose for contradiction that . Testing the equation for with we obtain
and thus is a minimizer for (2.3) and must have a unique zero by Step 1. However, clearly has a continuum of zeros. Therefore we can conclude that .
Step 3: has its unique zero either at or at . If we suppose for contradiction that the unique zero lies in the open interval , then we obtain the Euler-Lagrange equation (2.4) with boundary condition (2.5). By rescaling the minimizer suitably on we can achieve that the rescaled function is a -function on solving the equation (2.4) pointwise a.e. on . Hence, the rescaled function is a Neumann-eigenfunction with one interior zero, i.e., in contradiction to Step 2.
Now the claim of the lemma about the value of the minimum is immediate.
Step 4: ordering of . We are using the following rearrangement result of Hardy, Littlewood, Pólya . Let be non-negative and measurable on . If are the increasing rearrangements of , then . Moreover, if is strictly increasing, then equality holds if and only if . A similar statement holds for the decreasing rearrangements . Note, that the non-negativity of can be replaced by boundedness.
A simple corollary of the Hardy, Littlewood, Pólya inequality is the following: suppose is strictly increasing and both and are bounded. Then
with equality if and only if . The proof follows immediately from the observation that .
Let be strictly increasing on . Suppose for contradiction that and let be an eigenfunction corresponding to , which by (2.3) is also a minimizer of the variational problem in (2.3). We may assume to be non-negative, since is also a minimizer of the corresponding variational problem and is a simple eigenvalue. Let now be the decreasing rearrangement of on and note that . Since for the decreasing rearrangement we have , cf. Kawohl , we obtain by (2.7) applied to and the relations
Therefore , which satisfies , is also a minimizer of (2.3) and hence equality has to hold in (2.8). But since is strictly increasing, the sharp form of (2.7) implies that which by implies the contradiction that must be identically zero. Hence . Moreover, (2.8) shows that any non-negative minimizer for satisfies , i.e., is decreasing, and by using the differential equation for and the strict monotonicity of it is easy to see that in fact is strictly decreasing.
If is strictly decreasing on then a similar argument based on replacing by its increasing rearrangement shows that . ∎
3 Interface Problems
Let be the operator in (1.1) defined on the dense subset of . We investigate next the existence of eigenvalues of for the interface potentials (1.2) and (1.3). These examples fall into a larger class of potentials, namely , where for some but where may not be even in . Clearly, all solutions of are then
where are Bloch functions of , respectively. As decaying Bloch functions exist only in spectral gaps of , respectively, eigenvalues of can exist only within intersections of the gaps of and . Note the following additional information on the spectrum of , which for our purpose plays no further role: the essential spectrum of is the union of the essential spectra of and , cf. Korotyaev . As a result, no embedded eigenvalues of exist.
3.1 Point Spectrum for Interfaces Made of Even Potentials
coupled by the -matching conditions
As stated in Section 1, the functions are continuous, even and -periodic.
Based on the knowledge of the fundamental solutions in (2.1), (2.2) we conclude that an -integrable solution of (1.1) with (1.2) can only exist if lies in the intersection of the resolvent sets, i.e., in the intersection of the spectral gaps of and , i.e., if for some , where is the -th spectral gap of respectively.
For with some any localized eigenfunction of , therefore, has to be of the form
with and being periodic in . The functions are restrictions of either or in (2.1) with to the half-line respectively.
An important remark is that, due to the linearity of the problem, the matching conditions (3.2) together with an appropriate scaling are equivalent to
and the prime denotes differentiation in .
We determine existence of solutions to (3.4) via the intermediate value theorem and by monotonicity of the functions . The monotonicity then also implies uniqueness.
Within each gap and , the functions and are continuous functions of , which are strictly increasing and decreasing respectively.
Let us start with the proof for . Under the Prüfer transformation, cf. Coddington & Levinson 
the equation becomes
where the prime denotes differentiation in . Clearly, and are continuous functions of both variables and and since , the function is continuous in provided has no zero in the interior of . Note that if , then by evenness of and the reflection symmetry of the problem , the solution defined in (3.3) on could be extended to a solution on via . This solution would decay exponentially at both infinities and would, thus, be an eigenvalue of , which is impossible. Hence continuity of is proven.
Now let us prove the monotonicity. Due to the form of , see (3.3), we have
Define now . The function satisfies . Therefore,
Because , and due to the periodicity we have , where due to continuity the value is independent of .111In fact, it can be easily seen from Sturm oscillation theorem that , where is the index of the gap . Hence, . Using (3.5) and (3.6), we thus obtain
Because , we get and conclude that is strictly decreasing throughout . Therefore, is strictly increasing with respect to throughout .
In order to apply the intermediate value theorem and prove crossing of the graphs of and , we use their continuity within each gap and their limits as approaches a gap edge.
Let be one of the boundary-points of the spectral gaps of respectively. If corresponds to a Dirichlet-eigenvalue of on , then respectively, and if corresponds to a Neumann-eigenvalue of on then respectively.
We only consider the “” case. Let , be a given sequence. Due to (3.3) the functions have the form
where w.l.o.g. we may assume , which implies . On every compact subinterval the -norm of is uniformly bounded in and hence along a subsequence (again denoted by ) the functions converge in (and hence, by the differential equation (3.1) also in ) to a solution of with . Since this holds for every , the function is a bounded solution of on and therefore coincides with the bounded periodic Bloch function in (2.2). The convergence of is now obvious by the embedding into . ∎
To make the picture of the behavior of complete, it remains to determine their behavior at the lower end of the semi-infinite gap , i.e. as .
Let be bounded potentials (not necessarily even, periodic or continuous). Then as .
The proof is, as for Lemma 3.1, shown only for with the one for being completely analogous. We rescale the Bloch function so that . Note that this is possible if and only if , which we show to be true for all . Suppose that . Testing with over , we get
and, therefore, .
Let now for some , s.t. and , and define
Since , we need to determine the behavior of as . Using the Green’s function, we solve (3.9) to obtain
In order to estimate , (3.9) yields
which implies as . ∎
The behavior of the ratio functions for the two examples and is summarized in Figure 3. Note that Lemmas 2.2, 3.1, 3.2 and 3.3 imply the behavior only for . The rest in Figure 3 is obtained without a rigorous proof from numerical computations of the gap edge eigenfunctions.
By the intermediate value theorem and based on the behavior of , we now obtain the following theorem, which has already been observed by Korotyaev .