Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces

TL;DR

This paper derives asymptotic formulas for eigenvalues of Schrödinger operators on unbounded fractal spaces, extending spectral analysis techniques to complex geometries and verifying Bohr's formula under certain conditions.

Contribution

It establishes conditions for Bohr's formula to hold on metric measure spaces with cellular decompositions and verifies these on fractafolds and fractal fields, addressing spectral asymptotics for fractal-based operators.

Findings

Derived eigenvalue asymptotics for Schrödinger operators on unbounded fractals.

Verified Bohr's formula conditions for specific fractal spaces.

Partially answered spectral asymptotics question for harmonic oscillator on Sierpinski gasket.

Abstract

We establish an asymptotic formulas for the eigenvalue counting function of the Schr\"odinger operator $-\Delta +V$ for some unbounded potentials $V$ on several types of unbounded fractal spaces. We give sufficient conditions for Bohr's formula to hold on metric measure spaces which admit a cellular decomposition, and then verify these conditions for fractafolds and fractal fields based on nested fractals. In particular, we partially answer a question of Fan, Khandker, and Strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow-up of a Sierpinski gasket.