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?dinger operators , j≡ 1,2 in L ²((0,R)), 0<R≤∞, are exponentially close, that is, , 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?dinger operators. Received: 22 October 1999 / Accepted: 2 November 1999