The spectrum of a Schrödinger 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 mth 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 mth 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.
Mots-clés : Schrödinger operator, periodic structure, optimal design, spectral bandgap, Bravais lattices, rearrangement algorithm
@article{COCV_2019__25__A40_0,
author = {Kao, Chiu-Yen and Osting, Braxton},
title = {Extremal spectral gaps for periodic {Schr\"odinger} operators},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018029},
zbl = {1459.35296},
mrnumber = {4009413},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018029/}
}
TY - JOUR AU - Kao, Chiu-Yen AU - Osting, Braxton TI - Extremal spectral gaps for periodic Schrödinger operators JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018029/ DO - 10.1051/cocv/2018029 LA - en ID - COCV_2019__25__A40_0 ER -
%0 Journal Article %A Kao, Chiu-Yen %A Osting, Braxton %T Extremal spectral gaps for periodic Schrödinger operators %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018029/ %R 10.1051/cocv/2018029 %G en %F COCV_2019__25__A40_0
Kao, Chiu-Yen; Osting, Braxton. Extremal spectral gaps for periodic Schrödinger operators. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 40. doi : 10.1051/cocv/2018029. http://www.numdam.org/articles/10.1051/cocv/2018029/
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