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Eigenvalue
asymptotics and Bohr's formula for fractal Schrödinger
operators
Sze-Man Ngai and Wei Tang |
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Vol. 300 (2019), No. 1, 83–119
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Abstract |
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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. |
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Keywords
fractal, Schrödinger operator, Bohr's formula, Laplacian,
self-similar measure with overlaps
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Mathematical Subject Classification 2010
Primary: 28A80, 35J10
Secondary: 35J05, 35P20
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Milestones
Received: 27 November 2017
Revised: 30 July 2018
Accepted: 5 September 2018
Published: 20 July 2019
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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 |
