Complex powers of the Schrödinger operator with a regularly growing potential
type: Item
instance of: scholarly article
title: Complex powers of the Schrödinger operator with a regularly growing potential
zbMATH Open document ID: 0817.47065
DOI: 10.1007/BF01098025
author: S. Singh
published in: Journal of Soviet Mathematics
publication date: 1993-05-16T00:00:00Z
review text: Let \(A\) be the Schrödinger operator \[ Au(x)= -\Delta u(x)+ Q(x) u(x), \qquad x\in \mathbb{R}^ m, \] where the potential \(Q\) is assumed to be a nonnegative, infinitely differentiable and regularly increasing function at infinity, i.e., \[ c| x^ \alpha |\leq Q(x)\leq C| x|^ \alpha, \qquad \alpha>0, \] for \(| x|\) sufficiently large. Using the well established theory of pseudo-differential operators [see \textit{L. Hörmander}, Commun. Pure Appl. Math. 32, 359-443 (1979; Zbl 0388.47032)], the author finds the parametrix -- asymptotic expansion of the resolvent \((A- \lambda)^{-1}\) and then obtains the representation of the complex powers \(A^ z\) of the operators \(A\). In particular, he proves that the function \(z\mapsto A^ z\) can be meromorphically continued to the whole complex plain with poles of order not greater than two.
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 165458
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: pseudo-differential operators
zbMATH Keywords: parametrix
zbMATH Keywords: asymptotic expansion of the resolvent
zbMATH Keywords: complex powers
reviewed by: Vojislav Marić
MaRDI profile type: Publication
cites work: Q4157216
cites work: Q3897569
cites work: On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems
cites work: The weyl calculus of pseudo-differential operators
cites work: Proprietes spectrales d'operators pseduo-differentiels
cites work: Q5542418
full work available at URL: https://doi.org/10.1007/bf01098025
OpenAlex ID: W2012106765
description: scientific article; zbMATH DE number 165458