Eigenvalue asymptotics and Bohr's formula for fractal Schrodinger operators

Sze-Man Ngai Georgia Southern University and Hunan Normal University Wei Tang Hunan Normal University

Publications of CMSA of Harvard mathscidoc:1801.38001

For a fractal Schrodinger operator with a continuous potential and a coupling parameter, we obtain an analog of a semi-classical asymptotic formula for the number of bound states as the parameter tends to infinity. We also study Bohr's formula for Schrodinger operators on blowups of self-similar sets. For a Schrodinger operator defined by a fractal measure and a locally bounded potential that tends to infinity, we derive an analog of Bohr's formula under various assumptions. We demonstrate how these results 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 we call essentially of finite type.
Fractal; Schrodinger operator; Bohr's formula; Laplacian; self-similar measure with overlaps
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  title={Eigenvalue asymptotics and Bohr's formula for fractal Schrodinger operators},
  author={Sze-Man Ngai, and Wei Tang},
Sze-Man Ngai, and Wei Tang. Eigenvalue asymptotics and Bohr's formula for fractal Schrodinger operators. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180107040247842725873.
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