Localized Modes of the Linear Periodic Schrödinger Operator with a Nonlocal Perturbation

Abstract

We consider the existence of localized modes corresponding to eigenvalues of the periodic Schrödinger operator −∂ 2x + V (x) with an interface. The interface is modeled by a jump either in the value or the derivative of V (x) 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 C 1 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 R ± = ψ ± (0) ψ±(0) , where ψ ± 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 R ± 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.

The bulk-edge correspondence for continuous dislocated systems

“…This proves the bulk-edge correspondence for our model. It extends and generalizes parts of results of Korotyaev [72,73], Dohnal-Plum-Reichel [25] and Hempel-Kohlmann [58]. We compute the bulk/edge index on a few examples.…”

A variational approach to dislocation problems for periodic Schrödinger operators

“…Note that our calculations yield numbers which are exact up to finding the zeros of some transcendental functions. Related pictures can be found in [7] where a different numerical approach has been used.…”

Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons

“…Note that ω ∈ B r/2 (ω 0 ) is satisfied if ε > 0 is small enough. To estimate ∂ k ω W (•, ω 0 ) ∞ , k ∈ N, one proceeds by induction using (10), assumption (W1), and (11).…”

“…In [10] the case of real, periodic W ± , i.e. the one of two periodic conservative materials, was considered and eigenvalues were found by varying k and searching for zeros of R + (k) − R − (k).…”