Exact-dimensional property of density of states measure of Sturm Hamiltonian

TL;DR

This paper investigates the dimensional properties of the density of states measure for Sturm Hamiltonians with bounded type frequencies and strong coupling, revealing conditions under which it is exact-dimensional.

Contribution

It establishes that for certain parameters, the density of states measure is exact-dimensional, while in general it is not, advancing understanding of spectral measures in quantum models.

Findings

Density of states measure is exact upper and lower dimensional for bounded type frequencies and strong coupling.

In general, the measure is not exact-dimensional.

Provides conditions for exact-dimensionality in Sturm Hamiltonians.

Abstract

For frequency $\alpha$ of bounded type and coupling $\lambda>20$, we show that the density of states measure $\NN_{\alpha,\lambda}$ of the related Sturm Hamiltonian is exact upper and lower dimensional, however, in general it is not exact-dimensional.