Simon B.
A New Approach to Inverse Spectral Theory,
I. Fundamental Formalism
(64K, AMS-TeX)
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 $[0,a]$.