ar X iv : m at h / 99 10 08 9 v 1 [ m at h . SP ] 1 8 O ct 1 99 9 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 Schrödinger operators H j = − d 2 dx 2 + q j , j = 1, 2 in L 2 ((0, R)), 0 < R ≤ ∞, are exponentially close, that is, |m 1 (z) − m 2 (z)| = |z|→∞ O(e −2 Im(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 Schrödinger operators.