Localized Modes of the Linear Periodic Schr\"{o}dinger Operator with a Nonlocal Perturbation

TL;DR

This paper investigates localized eigenmodes in a periodic Schrödinger operator with an interface, analyzing how eigenvalues occur in spectral gaps and providing both theoretical and numerical insights into their properties.

Contribution

It introduces a novel approach to identify interface eigenvalues using a gluing problem and ratio functions, extending understanding of localized modes in nonlocal perturbations.

Findings

Eigenvalues occur only in spectral gaps.

The ratio functions are analyzed via Prüfer transformation.

Numerical results support the theoretical analysis.

Abstract

We consider the existence of localized modes corresponding to eigenvalues of the periodic Schr\"{o}dinger operator $-\partial_x^2+ V(x)$ 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 $R_\pm=\frac{\psi_\pm'(0)}{\psi_\pm(0)}$, where $\psi_\pm$ are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Pr\"{u}fer transformation. The limit values of $R_\pm$ 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.