Spectral properties of the Dirichlet operator $\sum _{i=1}^{d}(-\partial _i^2)^{s}$∑i=1d(−∂i2)s on domains in d-dimensional Euclidean space

Abstract

In this article, we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator \documentclass[12pt]{minimal}\begin{document}$\sum _{i=1}^{d}(-\partial _i^2)^{s}, \, s\in (0,1]$\end{document}∑i=1d(−∂i2)s,s∈(0,1] on an open and bounded subdomain \documentclass[12pt]{minimal}\begin{document}$\Omega \subset \mathbb {R}^d$\end{document}Ω⊂Rd 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 Schrödinger operator.