Spectral Properties of the Dirichlet Operator ${\sum }_{i=1}^d(-{\mathrm{\partial }}_i^2{)}^s$ on Domains in d-Dimensional Euclidean Space

In this article we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator $\sum_{i=1}^{d}(-\partial_i^2)^{s}, , s\in (1/2,1]$onanopenandboundedsubdomain$\Omega \subset $\mathbb{R}$^d$andpredictboundsonthesumofthefirst$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.