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

Abstract

We establish an asymptotic formulas for the eigenvalue counting function of the Schrödinger 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.

arXiv commentv2: 28 pages, 6 figures. referee comments included, typos corrected

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• Singularly continuous spectrum of a self-similar Laplacian on the half-line
• Wave equation on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials
• Regularized Laplacian determinants of self-similar fractals
• From non-symmetric particle systems to non-linear PDEs on fractals
• Fractatomic Physics: An Invitation with Atomic Stability and Rydberg States in Fractal Spaces