Defect Modes and Homogenization of Periodic Schr\"odinger Operators

TL;DR

This paper investigates how small localized perturbations affect the spectral properties of periodic Schr"odinger operators, leading to bifurcations of eigenvalues from band edges into spectral gaps, with analysis based on homogenization techniques.

Contribution

It introduces a detailed analysis of eigenvalue bifurcations in periodic Schr"odinger operators using homogenization, revealing the role of the effective mass matrix and localized potentials.

Findings

Eigenvalues bifurcate from spectral band edges into gaps for small perturbations.

The bifurcation behavior is governed by an associated homogenized Schr"odinger operator.

The effective mass matrix plays a key role in the bifurcation process.

Abstract

We consider the discrete eigenvalues of the operator $H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$ is localized on $\R^d,\ \ d\ge1$. For $\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schr\"odinger operator, $H_0 = -\Delta_\x+V(\x)$, into spectral gaps. The nature of the bifurcation depends on the homogenized Schr\"odinger operator $L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y)$. Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation.