Vol. 300, No. 1, 2019

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Eigenvalue asymptotics and Bohr's formula for fractal Schrödinger operators

Sze-Man Ngai and Wei Tang

Vol. 300 (2019), No. 1, 83–119
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
Authors
Sze-Man Ngai
College of Mathematics and Statistics
Hunan Normal University
Changsha
China
Department of Mathematical Sciences
Georgia Southern University
Statesboro, GA
United States
Wei Tang
Key Laboratory of High Performance Computing and Stochastic Information Processing
College of Mathematics and Statistics
Hunan Normal University
Changsha
China
College of Mathematical and Computational Science
Hunan First Normal University
Changsha
China