Extremal Spectral Gaps for Periodic Schr\"odinger Operators

TL;DR

This paper investigates the maximum spectral gap-to-midgap ratio for periodic Schr"odinger operators, proving existence of extremal potentials, characterizing their structure, and developing numerical methods to analyze their properties in one and two dimensions.

Contribution

It establishes the existence and structure of extremal potentials for spectral gaps and introduces efficient algorithms for their computation in multiple dimensions.

Findings

Optimal potentials are step-functions attaining bounds on exactly m intervals in 1D.

A rearrangement algorithm effectively computes extremal potentials numerically.

In 2D, a lattice of disks maximizes the first spectral gap in the infinite contrast limit.

Abstract

The spectrum of a Schr\"odinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the m-th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices.