On eigenfunction expansions and scattering theory
type: Item
instance of: scholarly article
title: On eigenfunction expansions and scattering theory
zbMATH Open document ID: 0793.47007
DOI: 10.1007/BF02571662
author: Thomas Poerschke
author: Günter Stolz
published in: Mathematische Zeitschrift
publication date: 1994-08-18T00:00:00Z
full work available at URL: https://eudml.org/doc/174491
review text: A systematic approach to expansions in generalized eigenfunctions for self-adjoint operators is presented. An abstract expansion theorem is proven, which allows a unified treatment of different types of eigenfunction expansions known for ordinary differential operators and in the spectral and scattering theory of Schrödinger operators, respectively. Previously this required different approaches. The application of this result to eigenfunction expansions which are related to scattering theory (IP-expansions) is studied in detail. The existence of such expansions is found as a consequence of the existence of wave operators. A very wide scope of scattering systems is treated (2- body or \(N\)-body, short range or long range, Stark effect-, Dirac- and two Hilbert space scattering).
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47A70
Mathematics Subject Classification ID: 81U10
zbMATH DE Number: 549760
zbMATH Keywords: IP-expansions
zbMATH Keywords: expansions in generalized eigenfunctions for self-adjoint operators
zbMATH Keywords: ordinary differential operators
zbMATH Keywords: spectral and scattering theory of Schrödinger operators
zbMATH Keywords: existence of wave operators
zbMATH Keywords: scattering systems
zbMATH Keywords: \(N\)-body scattering
zbMATH Keywords: short range scattering
zbMATH Keywords: long range scattering
zbMATH Keywords: Stark effect
zbMATH Keywords: Dirac- and two Hilbert space scattering
MaRDI profile type: Publication
cites work: Spectral and scattering theory of Schrödinger operators related to the Stark effect
cites work: Asymptotic completeness for N-body short-range quantum systems: A new proof
cites work: Q3752751
cites work: Q3705827
cites work: Fundamental solutions and eigenfunction expansions for Schrödinger operators. II. Eigenfunction expansions
cites work: Q3977205
cites work: Q3472542
cites work: On Ordinary Differential Equations of any Even Order and the Corresponding Eigenfunction Expansions
cites work: Expansions in generalized eigenfunctions of selfadjoint operators
cites work: Q4191141
cites work: Phase space analysis of simple scattering systems: extensions of some work of Enss
cites work: Schrödinger semigroups
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cites work: Spectral theory of ordinary differential operators
cites work: Long-Range Potential Scattering by Enss's Method in Two Hilbert Spaces
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description: scientific article; zbMATH DE number 549760