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