99-388 F. Gesztesy and B. Simon
On Local Borg-Marchenko Uniqueness Results (49K, LaTeX) 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 Weyl-Titchmarsh $m$-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 Borg-Marchenko uniqueness result (which is quickly recovered as a corollary to our proof). Moreover, we extend the local uniqueness result to matrix-valued Schr\"odinger operators.

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