  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.
