 99388 F. Gesztesy and B. Simon
 On Local BorgMarchenko Uniqueness Results
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Oct 18, 99

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Abstract. We provide a new short proof of the following fact, first proved by one of us in 1998: If two WeylTitchmarsh $m$functions, $m_j(z)$, of two Schr\"odinger operators $H_j = \f{d^2}{dx^2} + q_j$, $j=1,2$ in $L^2 ((0,R))$, $0<R\leq \infty$, are exponentially close, that is, $m_1(z) m_2(z) \underset{z\to\infty}{=} O(e^{2\Ima (z^{1/2})a})$, $0<a<R$, then $q_1 = q_2$ a.e.~on $[0,a]$. The result applies to any boundary conditions at $x=0$ and
$x=R$ and should be considered a local version of the celebrated BorgMarchenko
uniqueness result (which is quickly recovered as a corollary to our proof).
Moreover, we extend the local uniqueness result to matrixvalued Schr\"odinger
operators.
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