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 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) = -κ- \int_0^b A(α) e^{-2ακ}\, dα+ O(e^{-(2b-\varepsilon)κ}). 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 α} + \int_0^αA(β, x) A(α-β, x)\, dβ. 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].