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 $m(-\kappa^2) = -\kappa -\int_0^b A(\alpha) e^{-2\alpha\kappa} d\alpha+O(e^{-(2b-\varepsilon)\kappa})$. $A$ 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 $[0,a]$.


Barry Simon