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

Abstract

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].