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A new approach to inverse spectral theory, I. Fundamental formalism

Simon, Barry (1999) A new approach to inverse spectral theory, I. Fundamental formalism. Annals of Mathematics, 150 (3). pp. 1029-1057. ISSN 1012-2443 .

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We present a new approach (distinct from Gel′fand-Levitan) to the theorem of Borg-Marchenko that the m-function (equivalently, spectral measure) for a finite interval or half-line Schr¨odinger operator determines the potential. Our approach is an analog of the continued fraction approach for the moment problem. We prove there is a representation for the m-function m(−κ2) = [eqation]. A on [0, a] is a function of q on [0, a] and vice-versa. A key role is played by a differential equation that A obeys after allowing x-dependence: [equation] Among our new results are necessary and sufficient conditions on the m-functions for potentials q1 and q2 for q1 to equal q2 on [0, a].

Item Type:Article
Simon, Barry0000-0003-2561-8539
Additional Information:(Received December 30, 1997) I thank P. Deift, I. Gel′fand, R. Killip, and especially F. Gesztesy, for useful comments, and M. Ben-Artzi for the hospitality of Hebrew University where part of this work was done.
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Deposited On:29 Sep 2005
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