#### 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