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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 Schrödinger 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(−κ^2) = -K-ʃ^b_0 A(ɑ)e^(-2ɑk)dɑ + O(e^-(2b-є)^k). A on [0, ɑ] is a function of q on [0, ɑ] and vice-versa. A key role is played by a differential equation that A obeys after allowing x-dependence: ∂A/∂x = ∂A/∂ɑ + ʃ^ɑ_0 a(βX)A(ɑ - β,x) dβ. 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, ɑ].

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Simon, Barry0000-0003-2561-8539
Additional Information:© 1999 Annals of Mathematics. 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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Official Citation:A New Approach to Inverse Spectral Theory, I. Fundamental Formalism Author(s): Barry Simon Annals of Mathematics, Second Series, Vol. 150, No. 3 (Nov., 1999), pp. 1029-1057 Published by: Annals of Mathematics Stable URL:
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ID Code:769
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Deposited On:29 Sep 2005
Last Modified:15 Mar 2018 03:39

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