On Local Borg-Marchenko Uniqueness Results

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 $H_j=-\text{f}{d}^{2}d{x}^2+q_j$, j=1,2 in $L^2((0,R))$, $0<R\leq \infty$,areexponentiallyclose,thatis,$|m_1(z)- m_2(z)| \underset{|z|\to\infty}{=} O(e^{-2\Ima (z^{1/2})a})$,0<a<R,then$q_1 = q_2$a\mathrm{.}e\mathrm{.}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.