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SINGULAR SCHRÖDINGER OPERATORS IN ONE DIMENSION

Part of: Ordinary differential operators Boundary value problems

Published online by Cambridge University Press:  12 April 2012

E. B. Davies
E. B. Davies*
Affiliation:
Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, U.K. (email: E.Brian.Davies@kcl.ac.uk)

Abstract

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We consider a class of singular Schrödinger operators H that act in L2(0,∞), each of which is constructed from a positive function ϕ on (0,∞). Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of ϕ. For a large class of H that have discrete spectrum, we prove that the eigenvalue asymptotics of H does not depend on rapid oscillations of ϕ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.

MSC classification

Secondary: 34B27: Green functions 34L05: General spectral theory 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions 34L25: Scattering theory, inverse scattering

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 141 - 159
Copyright
Copyright © University College London 2012

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