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(-\kappa^2) = -\kappa - \int_0^b A(\alpha) e^{-2\alpha\kappa}\, d\alpha + O(e^{-(2b-\varepsilon)\kappa}). 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: \frac{\partial A}{\partial x} = \frac{\partial A}{\partial \alpha} + \int_0^\alpha A(\beta, x) A(\alpha -\beta, x)\, d\beta. Among our new results are necessary and sufficient conditions on the m-functions for potentials q_1 and q_2 for q_1 to equal q_2 on [0,a].