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A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure

Geszetsy, Fritz and Simon, Barry (2000) A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure. Annals of Mathematics, 152 (2). pp. 593-643. ISSN 0003-486X. https://resolver.caltech.edu/CaltechAUTHORS:20111130-133257722

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Abstract

We continue the study of the A-amplitude associated to a half-line Schrödinger operator, - d^2/dx^2 + q in L^2((0,b)), b ≤ ∞ A is related to the Weyl-Titchmarsh m-function via m(-k^2) = -k- ʃ^a_0 A(α)e^(-2αk) dα+O(e^(-(2α-Є)k)) for all Є > 0. We discuss five issues here. First, we extend the theory to general q in L^1((0,α)) for all a, including q's which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure p: A(α) = -2 ^ʃ∞_(-∞)λ^(-1/2) sin (2α√λ) dp(λ) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b < ∞. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.2307/2661393DOIUNSPECIFIED
http://www.jstor.org/stable/2661393PublisherUNSPECIFIED
ORCID:
AuthorORCID
Simon, Barry0000-0003-2561-8539
Additional Information:© 2000 Princeton University. Received June 10, 1999. This material is based upon work supported by the National Science Foundation under Grant No. DMS-9707661. The government has certain rights in this material.
Funders:
Funding AgencyGrant Number
NSFDMS-9707661
Subject Keywords:Inverse spectral theory, Weyl-Titchmarsh m-function, spectral measure.
Issue or Number:2
Classification Code:1991 MSC: Primary: 34A55, 34B20; Secondary: 34L05, 47A10
Record Number:CaltechAUTHORS:20111130-133257722
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20111130-133257722
Official Citation:Fritz Gesztesy and Barry Simon: A new approach to inverse spectral theory. II: General real potentials and the connection to the spectral measure
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28253
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:30 Nov 2011 22:45
Last Modified:03 Oct 2019 03:30

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