Sharp bounds on the number of scattering poles in even-dimensional spaces
type: Item
instance of: scholarly article
title: Sharp bounds on the number of scattering poles in even-dimensional spaces
zbMATH Open document ID: 0813.35075
DOI: 10.1215/S0012-7094-94-07401-2
author: Georgi Vodev
published in: Duke Mathematical Journal
publication date: 1995-05-29T00:00:00Z
review text: Scattering poles or resonances can be defined as the poles of the meromorphic continuation of the cut-off resolvent of say a Laplace operator under perturbation when such a continuation exists. For odd space dimension \((\geq 3)\) the cut off resolvent extends meromorphically to the entire complex plane while in the even dimensional case a meromorphic extension is possible only in the Riemann logarithmic surface \(\Lambda = \{Z : - \infty < \arg z < \infty\}\). Upper bounds on the member of scattering poles in odd dimensions are derived using Jensen's inequality on the zeros of holomorphic functions. In the odd case, given certain localization hypotheses on the operator \(G\) with resolvent \(R(Z)\), it is shown that the operator \(R_ \chi (Z) = \chi R(Z) \chi,\chi=1\) on \(B_ \rho=\{x:| x|\leq\rho\}\), extends meromorphically to \(\Lambda\) with poles \(\lambda_ j\). The scattering poles in \(\Lambda_ r = \{\Lambda_ n | z | < r\}\), with multiplicity, are among the zeros of a function \(h_ r(Z) = \text{det} (I-S_ r (Z))\), holomorphic in \(\Lambda_ r\), where \(S_ r(Z)\) is a function defined in a fairly complex way from \(R_ \chi (Z)\). Again the author establishes an analogue of Jensen's inequality for functions holomorphic on the Riemann logarithmic surface using a generalised version of the classical Carleman theorem. As a consequence he is able to determine that the counting function \(N(r,a)\) of the scattering poles of \(G\) satisfies the bound \[ N(r,a) \leq Ca \bigl( \varphi (Cr) + (j \circ ga)^ n \bigr), \quad \forall r,\;a>1, \] with a constant \(C>0\) independent of \(r\) and \(a\), where \(\varphi\) is a function whose growth properties are related to the smoothing properties of \(G\).
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 47A40
zbMATH DE Number: 624869
zbMATH Keywords: resonances
zbMATH Keywords: Riemann logarithmic surface
zbMATH Keywords: Jensen's inequality
MaRDI profile type: Publication
cites work: A polynomial bound on the number of the scattering poles for a potential in even dimensional spaces IRn
cites work: Q3739718
cites work: Q5579836
cites work: Q5531193
cites work: Polynomial bound on the number of scattering poles
cites work: Q3756685
cites work: Weyl asymptotics for the phase in obstacle scattering
cites work: Complex Scaling and the Distribution of Scattering Poles
cites work: Lower bounds on the number of scattering poles
cites work: Lower bounds on the number of scattering poles. II
cites work: Q3999258
cites work: Sharp polynomial bounds on the number of scattering poles for metric perturbations of the Laplacian in \(\mathbb{R}{}^ n\)
cites work: Sharp bounds on the number of scattering poles for perturbations of the Laplacian
cites work: Sharp polynomial bounds on the number of scattering poles of radial potentials
cites work: Sharp polynomial bounds on the number of scattering poles
full work available at URL: s0012-7094-94-07401-2
OpenAlex ID: W1497267102
description: scientific article; zbMATH DE number 624869