 97645 Simon B.
 A New Approach to Inverse Spectral Theory,
I. Fundamental Formalism
(64K, AMSTeX)
Dec 23, 97

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Abstract. We present a new approach (distinct from Gel'fandLevitan) to
the theorem of BorgMarchenko that the $m$function
(equivalently, spectral measure) for a finite interval or
halfline 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 viceversa.
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 $[0,a]$.
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