Maximal Length Spectral Gaps for Periodic Schroedinger Operators
Wednesday, April 26, 2017 - 10:30am - 11:30am
The spectrum of a Schroedinger operator with periodic potential generally consists of bands and gaps. In this talk, for fixed m, I'll 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 found 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. I'll also compare our results with the analogous problem for the Helmholtz operator. This is joint work with Chiu-Yen Kao.