Spectral Properties of the Dirichlet Operator $\sum_{i=1}^d (-\partial_i^2)^s$ on Domains in d-Dimensional Euclidean Space

TL;DR

This paper studies the eigenvalue distribution of a fractional Dirichlet operator on bounded domains, providing bounds and asymptotic formulas, and connects these results to semi-classical quantum mechanics via coherent states.

Contribution

It offers new bounds and asymptotic estimates for eigenvalues of fractional Dirichlet operators and links these to semi-classical quantum mechanics methods.

Findings

Derived bounds on eigenvalue sums and counting functions.

Established asymptotic formulas for eigenvalue distributions.

Connected spectral properties to semi-classical quantum mechanics.

Abstract

In this article we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator $\sum_{i=1}^{d}(-\partial_i^2)^{s}, \, s\in (1/2,1]$ on an open and bounded subdomain $\Omega \subset \mathbb{R}^d$ and predict bounds on the sum of the first $N$ eigenvalues, the counting function, the Riesz means and the trace of the heat kernel. Moreover, utilizing the connection of coherent states to the semi-classical approach of Quantum Mechanics we determine the sum for moments of eigenvalues of the associated Schr\"{o}dinger operator.