Eigenvalue asymptotics and Bohr's formula for fractal Schrödinger operators

Sze-Man Ngai and Wei Tang

Vol. 300 (2019), No. 1, 83–119

DOI: 10.2140/pjm.2019.300.83

Abstract

For a Schrödinger operator defined by a fractal measure with a continuous potential and a coupling parameter, we obtain an analog of a semiclassical asymptotic formula for the number of bound states as the parameter tends to infinity. We also study Bohr’s formula for fractal Schrödinger operators on blowups of self-similar sets. For a locally bounded potential that tends to infinity, we derive an analog of Bohr’s formula under various assumptions. We demonstrate how this result can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, a family of convolutions of Cantor-type measures, and a family of measures that are essentially of finite type.

Keywords

fractal, Schrödinger operator, Bohr's formula, Laplacian, self-similar measure with overlaps

Mathematical Subject Classification 2010

Primary: 28A80, 35J10

Secondary: 35J05, 35P20

Milestones

Received: 27 November 2017

Revised: 30 July 2018

Accepted: 5 September 2018

Published: 20 July 2019