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) = -K-ʃ^b_0 A(ɑ)e^(-2ɑk)dɑ + O(e^-(2b-є)^k). A on [0, ɑ] is a function of q on [0, ɑ] and vice-versa. A key role is played by a differential equation that A obeys after allowing x-dependence: ∂A/∂x = ∂A/∂ɑ + ʃ^ɑ_0 a(βX)A(ɑ - β,x) dβ. 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, ɑ].