Spectral Properties of the Dirichlet Operator $∑_i=1^d (-∂_i^2)^s$ on Domains in d-Dimensional Euclidean Space

In this article we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator $∑_i=1^d(-∂_i^2)^s, \, s∈ (1/2,1]$ on an open and bounded subdomain $Ω ⊂ \mathbbR^d$ 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 Schrödinger operator.