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In this article we investigate the distribution of eigenvalues of theDirichlet 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 countingfunction, the Riesz means and the trace of the heat kernel. Moreover, utilizingthe connection of coherent states to the semi-classical approach of QuantumMechanics we determine the sum for moments of eigenvalues of the associatedSchr\"{o}dinger operator.

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