Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes
Let denote a periodic function on the real line. The Schrödinger operator, , has spectrum equal to the union of closed real intervals separated by open spectral gaps. In this article we study the bifurcation of discrete eigenvalues (point spectrum) into the spectral gaps for the operator , where is spatially localized and highly oscillatory in the sense that its Fourier transform, is concentrated at high frequencies. Our assumptions imply that may be pointwise large but is small in an average sense. For the special case where with smooth, real-valued, localized in , and periodic or almost periodic in , the bifurcating eigenvalues are at a distance of order from the lower edge of the spectral gap. We obtain the leading order asymptotics of the bifurcating eigenvalues and eigenfunctions. Consider the spectral band () of . Underlying this bifurcation is an effective Hamiltonian associated with the lower spectral band edge: where is the Dirac distribution, and effective-medium parameters are explicit and independent of . The potentials we consider are a natural model for wave propagation in a medium with localized, high-contrast and rapid fluctuations in material parameters about a background periodic medium.
Let denote a one-periodic function on the real line:
The Schrödinger operator,
has spectrum equal to the union of closed real intervals (spectral bands) separated by open spectral gaps. It is known that a spatially localized and small perturbation of , say , where , induces the bifurcation of discrete eigenvalues (point spectrum) from the edge of the continuous spectrum (zero energy) into the spectral gaps at a distance of order from the edge of spectral bands; see, e.g. [23, 15, 9]. In this article we study the bifurcation of discrete spectrum for the operator , where is localized in space and such that its Fourier transform is concentrated at high frequencies. A special case we consider is: , where is smooth, real-valued, localized in and periodic or almost periodic in . In this case, tends to zero weakly but not strongly.
Our motivation for considering such potentials is the wide interest in wave propagation in media (i) whose material properties vary rapidly on the scale of a characteristic wavelength of propagating waves and (ii) whose material contrasts are large. We model rapid variation by assuming that the leading-order component of the perturbation is supported at ever higher frequencies (asymptotically as ), and we allow for high contrast media by not requiring smallness on the norm of . Such potentials have some of the important features of high contrast micro- and nano-structures (see e.g. , ) and, more generally, wave-guiding or confining media with a multiple scale structure.
We obtain detailed leading order asymptotics of bifurcating eigenvalues and their associated eigenfunctions, with error bounds, in the limit as tends to zero. The present article generalizes our earlier work [9, 10] for the case (homogeneous background medium) and for , where is taken to be non-trivial and periodic and is small and localized in space.
Standard homogenization theory (averaging, in this case), which often applies in situations of strong scale-separation, does not capture the key bifurcation phenomenon. This was discussed in detail in . Underlying the bifurcation is an effective Dirac distribution potential well; the bifurcation at the lower edge of the spectral band of () is governed by an effective Hamiltonian . Here, are independent of and are given explicitly in terms of , . This reveals the leading-order location of the bifurcating eigenvalue at a distance from the spectral band edge.
1.1 Discussion of results
To describe our results in greater detail, we first present a short review of the spectral theory of ; see, for example, [12, 20]. The spectrum is determined by the family of self-adjoint pseudo-periodic eigenvalue problems, parametrized by the quasi-momentum :
listed with multiplicity, and corresponding pseudo-periodic normalized eigenfunctions:
The spectral band is given by . The spectrum of is given by: . Since the boundary condition (1.4) is invariant with respect to , the functions can be extended to all as periodic functions of . The minima and maxima of occur at ; see Figure 1. If and is a spectral band endpoint, bordering on a spectral gap, then is a simple pseudo-periodic eigenvalue, , and is either strictly positive or strictly negative; see Lemma 2.2.
Consider now the perturbed operator , where is sufficiently localized in . By Weyl’s theorem on the stability of the essential spectrum, one has . Therefore, the effect of a localized perturbation is to possibly introduce discrete eigenvalues into the spectral gaps. Note that does not have discrete eigenvalues embedded in its continuous spectrum; see , .
Theorem 3.1 () and Theorem 3.2 ( non-trivial periodic) are our main results on bifurcation of discrete eigenvalues of from the left (lower) band edge into spectral gaps of . They apply to spatially localized and spectrally supported at ever higher frequencies as (hence weakly convergent as ). In this introduction, we state for simplicity the results for the particular case of periodic and a two-scale function (spatially localized on on the slow scale and almost periodic on the fast scale) of the form:
where the frequencies satisfy the nonclustering assumptions:
for some fixed . The constraint that be real-valued implies: and . The particular case corresponds to being periodic.
Let , denote the lower edge of the spectral band and assume that this point borders a spectral gap; see the left panel of Figure 1. Assume is of the form (1.7) and is sufficiently smooth and decays sufficiently rapidly as and ; see Lemma C.1 and Theorem 3.2.
Let and denote the effective-medium parameters
Then, there exist constants and , such that for all the following holds:
has a simple discrete eigenvalue, (see the right panel in Figure 1);
with corresponding localized eigenfunction, :
Here, is the unique eigenvalue (simple) of the effective operator
where denotes the Dirac delta mass at , and is its corresponding eigenfunction (unique up to a multiplicative constant).
Theorem 1.1 applies to the special case: . Indeed, the spectrum of consists of a semi-infinite interval, , the union of intersecting bands with no positive length gaps. The only band-edge is located at , where we have: , , for all and , and therefore
Thus we recover the result of , where it was shown that the bifurcation at the lower edge of the continuous spectrum of is governed by the Hamiltonian corresponding to a small effective potential well on the slow length-scale:
Remark 1.4 (Examples of , not of standard two-scale type).
As mentioned earlier, our results apply in more general situations than the two-scale perturbation presented above. The assumptions of Theorems 3.1 and 3.2 imply that the leading-order component of the perturbation is supported at ever higher frequencies, asymptotically as . The main difficulty in a specific situation is to check assumption (H2) in Theorem 3.1 (resp. (H2’) in Theorem 3.2) the existence of effective coupling coefficient, .
with, for example, the assumptions and . This allows for dependence of on two-, three- etc. scales.
One further non-standard example to which our theorems apply is obtained by taking
where for small ( sufficiently large) and decaying sufficiently rapidly as . In this case,
1.2 Motivation, method of proof and relation to previous work
In  and in  the case where , with is considered under different hypotheses. Our analysis in  allows for almost periodic dependence in the fast-scale variable, i.e. potentials of the type displayed in (1.7). In this work we obtain details about eigenvalue asymptotics, and far more, by deriving asymptotics of the transmission coefficient, , that are valid uniformly for and in a complex neighborhood of zero energy. This enables us to control the spectral measure of , , leading to detailed dispersive energy transport information (time-decay estimates) in addition to results on eigenvalue-bifurcation.
The subtlety in this analysis stems from the behavior in a neighborhood of . Indeed, bounded away from , uniformly; see . The heart of the matter is a proof that
can be made to converge to zero as uniformly on (and in a complex neighborhood of ) for the specific choice ; see Remark 1.2. Since is a small potential well, classical results  for the operator apply, and we conclude that and consequently have a simple pole of order on the positive imaginary axis, from which the existence of a negative discrete eigenvalue, , of order is an immediate consequence. More precisely, the asymptotic behavior of the eigenvalue corresponding to the small potential well, and therefore to the original oscillatory potential, is predicted by the Schrödinger operator with Dirac distribution potential with negative mass (see , consistently with [23, 5]):
Since perturbations of the periodic Hamiltonian by weak potentials are also known to generate discrete eigenvalues, seeking an extension of the results in  to the case of a non-trivial and periodic background was a natural motivation for the current article.
then an eigenvalue of order bifurcates from the edge of the spectral band of the unperturbed operator . If and , this bifurcation is from the lower edge of the band, while if and the bifurcation is from the upper edge of the band.
Consistent with the case , in this work we prove that the spectral properties of the Hamiltonian localized near the band edge are related to those of an effective Hamiltonian
Upon rescaling by gives the operator , displayed in (1.12).
In contrast to the case of a multiplicatively small perturbation, the eigenvalue bifurcations of are shown in the present work to occur only from the lower band edge into the spectral gap below it. The mathematical reason for this is that the bifurcation phenomena we study is an effect that occurs at second order in . Making this effect explicit requires iteration of our formulation of the eigenvalue problem, leading to terms which are quadratic in . As in the case , the dominant (resonant / non-oscillatory) contribution has the distinguished sign of a potential well; see Remark 1.3. This result was also observed in [1, Corollary 2.1].
Non-oscillatory perturbations of Schrödinger operators with periodic background have been considered in a number of other works; see [8, 15, 16, 6]. For the acoustic and Maxwell operators see [13, 14]. Finally, Borisov and Gadylshin [1, 3, 4] obtained results which apply to our situation provided the perturbation is a two-scale potential and has compact support (neither hypothesis is required in our analysis). In , one-dimensional divergence-form operators are treated.
In two space dimensions, the operator , where and is a localized potential well, has a discrete negative eigenvalue of order ; see, for example, [23, 19]. In , Borisov proves that eigenvalues of the operator , where is periodic on , bifurcate from the edges of the continuous spectrum at a distance . It is natural to
Conjecture: In two space dimensions , where is periodic on and is spatially localized and concentrated at ever higher frequencies as as in (1.7), spawns eigenvalues from its lower spectral band edges into open gaps at a distance .
Finally, we remark on our method of analysis. We transform the eigenvalue problem using the natural basis of eigenfunctions for the unperturbed operator and study the eigenvalue problem in (quasi-) momentum space. The momentum space formulation is natural in that one can very systematically pinpoint the key resonant (non-oscillatory) terms which control the limit. Using this approach one sees clearly how to treat oscillatory perturbing potentials which are far more general than a prescribed multiscale type (two-scale, three-scale etc.). We explicitly, via localization to energies near the bifurcation point and rescaling, re-express the Schrödinger eigenvalue problem with rapidly oscillatory coefficients as an approximately equivalent eigenvalue problem for an effective Schrödinger operator, , with coefficients which do not oscillate rapidly. This effective Schrödinger Hamiltonian is determined by key constants and , which have natural physical meanings (inverse effective mass and effective potential well couple parameter, respectively).
The main tool for re-expressing the eigenvalue problem is careful integration by parts, which exploits oscillations of non-resonant (“irrelevant”) terms to show that they are small in norm. Resonant (non-oscillatory) terms cannot be transformed to terms of high order in the small parameter and it is these terms that contribute to the effective operator, . Thus our approach is somewhat akin to that taken in Hamiltonian normal form theory and the method of averaging. See also .
1.3 Outline of the paper
In Section 2 we present background material concerning spectral properties of Schrödinger operators with periodic potentials defined on . In Section 3 we give precise technical statements of our main results: Theorem 3.1 and Theorem 3.2. Section 4 reviews general technical results on a class of band-limited Schrödinger operators, derived in , and applied in Sections 6 and 7. The strategy of the proof is explained in Section 5. Appendix A gives detailed proofs of bounds used in Section 7. Appendix B summarizes and proves bounds relating to the Floquet-Bloch states used in Section 7. Finally, Appendix C has a detailed analysis and calculation of the effective potential for the particular case of the localized and oscillatory (almost periodic) potential , defined in (1.7).
1.4 Definitions and notation
We denote by a constant, which does not depend on the small parameter, . It may depend on norms of and , which are assumed finite. is a constant depending on the parameters , , . We write if , and if and .
The methods of this paper employ spectral localization relative to the background operator , where is one-periodic. For the case, , we use the classical Fourier transform and for a non-trivial periodic potential, we use the spectral decomposition of in terms of Floquet-Bloch states; see Section 1 and Section 2 below. The notations and conventions we use are similar to those used in .
For , the Fourier transform and its inverse are given by
and denote the Gelfand-Bloch transform and its inverse, defined in (2.4) and (2.11) respectively. We use the following notation for the Gelfand-Bloch transform of a function: ; see section 2. Note that we will also use the notation in Section 7 to represent the projection of onto a particular Bloch function , for fixed .
and are the characteristic functions defined for a parameter by
We also use the notation
is the space of functions such that , endowed with the norm
is the space of functions such that for , endowed with the norm
Acknowledgements: The authors thank the referees and editor for their careful reading of our article and for their suggestions. I.V. and M.I.W. acknowledge the partial support of U.S. National Science Foundation under U.S. NSF Grants DMS-10-08855, DMS-1412560, the Columbia Optics and Quantum Electronics IGERT NSF Grant DGE-1069420 and NSF EMSW21- RTG: Numerical Mathematics for Scientific Computing. Part of this research was carried out while V.D. was the Chu Assistant Professor of Applied Mathematics at Columbia University.
2 Mathematical background
In this section we provide further mathematical background by summarizing basic results on the spectral theory of Schrödinger operators with periodic potentials defined on . Specifically, in Section 2.1 we discuss more detailed aspects of Floquet-Bloch theory, the spectral theory of periodic Schrödinger operators, and in Section 2.2 we introduce the Gelfand-Bloch transform and discuss its properties. For a detailed discussion, see for example, [12, 20, 18].
2.1 Floquet-Bloch theory
For continuous and one-periodic, consider the family of pseudo-periodic eigenvalue problems
parametrized by , the Brillouin zone. Setting , this is equivalent to the family of periodic boundary value problems:
for each .
The solutions may be chosen so that is, for each fixed , a complete orthonormal set in . It can be shown that the set of Floquet-Bloch states is complete in , i.e. for any ,
Recall that the spectrum of is the union of the spectral bands:
We say there is a spectral gap between the and bands if
Our analysis of eigenvalue-bifurcation from the band edge into a spectral gap, requires detailed properties of , e.g. regularity, near its edges. These are summarized in the following two results; see, for example,  and .
Assume is an endpoint of a spectral band of , which borders on a spectral gap. Then and the following results hold:
is a simple eigenvalue of the eigenvalue problem (2.1).
even: corresponds to the left (lowermost) endpoint of the band,
corresponds to the right (uppermost) endpoint.
odd: corresponds to the right (uppermost) endpoint of the band,
corresponds to the left (lowermost) endpoint.
even: , ;
odd: , ;
For real, consider the Floquet-Bloch eigenpair . Assume , is a simple eigenvalue. Then, there are analytic mappings , with normalized, defined for in a sufficiently small complex neighborhood of .
There exists such that for any and ,
2.2 The Gelfand-Bloch transform
Let , the Schwartz space. We introduce the Gelfand-Bloch transform or , as follows
Note the following properties of . For any , one has
Furthermore, for any we have . Therefore, for any sufficiently regular one-periodic function ,
Now, recall Poisson summation formula:
One deduces the following identity for :
This yields in particular the following formula for the Bloch transform of a product of two functions.
The Bloch transform of a product of two functions can be written as a “Bloch convolution”:
Note that for , the integrand is evaluated using (2.6).
Introduce the operator :
One can check that is the inverse of , .
For any Floquet-Bloch mode,
we have, thanks to (2.9),
By completeness of the , we deduce
The above definitions and identities extend by density to , and one has in particular for any ,
It will be natural to measure (Sobolev) regularity in terms of the decay properties of a function’s Floquet-Bloch coefficients. Thus we introduce the norm:
is isomorphic to for . Moreover, there exist positive constants , such that for all , we have
3 Bifurcation of defect states into gaps; main results
In this section we state our main results on the eigenvalue problem
where is one-periodic and a real-valued, localized at high frequencies and decreasing at infinity (precise hypotheses are specified below).
Consider first the case where . The following result extends Corollary 3.7 of  to a larger class of localized and oscillatory potentials, .
Assume that is real-valued and satisfies the following, for sufficiently small: