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 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(−κ2) = −κ− ∫ b 0 A(α)e−2ακ dα + O(e−(2b−ε)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: