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

Agapitos N. Hatzinikitas

doi:10.1063/1.4823481

paper

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

arXiv:1301.4806 · doi:10.1063/1.4823481

Abstract

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 $Ω \subset R^{d}$ 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 Schrödinger operator.

16 pages

Spectral Properties of the Dirichlet Operator ∑i=1d​(−∂i2​)s on Domains in d-Dimensional Euclidean Space

Abstract

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