An extension of the Ikehara Tauberian theorem and its application
type: Item
instance of: scholarly article
title: An extension of the Ikehara Tauberian theorem and its application
zbMATH Open document ID: 0857.35096
DOI: 10.1007/BF00114420
author: Junichi Aramaki
published in: Acta Mathematica Hungarica
publication date: 1997-03-09T00:00:00Z
review text: In this article [which is the fifth of a series devoted by the author to this thema, see e.g. J. Funct. Anal. 87, No. 2, 294-320 (1989; Zbl 0697.47011)] the author presents an extension of the Ikehara's Tauberian theorem relating to a selfadjoint operator with compact resolvent of the properties of the meromorphic function \(s\mapsto\text{Tr } P^{-s}\) to the asymptotic behavior, when \(\lambda\) is large, of the counting function \(N_P(\lambda)\) of the eigenvalues of \(P\) which are less than \(\lambda\). This is applied to the case when \(P\) takes the form \[ P=-\Delta_x-\Delta_y+ (1+|x|^2)^p|y|^{2q}, \] on \(\mathbb{R}^n_x\times \mathbb{R}^m_y\). This example was previously analyzed by \textit{D. Robert} [J. Math. Pures Appl. 61, 275-300 (1982; Zbl 0511.35069)] but without the same control on the remainder term.
reviewed by: Bernard Helffer
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 40E05
zbMATH DE Number: 897186
zbMATH Keywords: counting function
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00114420
OpenAlex ID: W2027240367
description: scientific article; zbMATH DE number 897186