Eigenvalue asymptotics and Bohr's formula for fractal Schrödinger operators

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.