Spectral properties of the Dirichlet operator $\sum _{i=1}d}(-\partial _i arrow up )s}$?i=1d( partial differential i2)s on domains in d-dimensional Euclidean space

In this article, we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator \documentclass[12pt]{minimal}\begin{document}$\sum _{i=1}d}(-\partial _i arrow up )s}, \, s\in (0,1]$\end{document} capital sigma i=1d(- partial differential i2)s,s eta (0,1 ] on an open and...

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Bibliographic Details
Published in:Journal of mathematical physics 2013-01, Vol.54 (10)
Main Author: Hatzinikitas, Agapitos N
Format: Article
Language:English
Subjects:
Counting
Dirichlet problem
Eigenvalues
Functions (mathematics)
Mathematical analysis
Operators
Schroedinger equation
Spectra
Online Access:Get full text
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Summary:In this article, we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator \documentclass[12pt]{minimal}\begin{document}$\sum _{i=1}d}(-\partial _i arrow up )s}, \, s\in (0,1]$\end{document} capital sigma i=1d(- partial differential i2)s,s eta (0,1 ] on an open and bounded subdomain \documentclass[12pt]{minimal}\begin{document}$\Omega \subset \mathbb {R} delta $\end{document} Omega sub(Rd 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 Schrodinger operator.)
ISSN:0022-2488
DOI:10.1063/1.4823481