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

• Autores: Barry Simon
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 150, Nº 3, 1999, págs. 1029-1057
• Idioma: inglés
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• Resumen

  • 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¿Nodinger 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(.¿È2) = .¿È . Rb 0 A(¿¿)e.2¿¿¿È d¿¿ + O(e.(2b.¿Ã)¿È). 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:

    ¿ÝA ¿Ýx = ¿ÝA ¿Ý¿¿ + Z ¿¿ 0 A(¿À, x)A(¿¿ . ¿À, x) d¿À.

    Among our new results are necessary and sufficient conditions on the mfunctions for potentials q1 and q2 for q1 to equal q2 on [0, a].