On Local Borg–Marchenko Uniqueness Results

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, mj(z), of two Schrödinger operators , j≡ 1,2 in L2((0,R)), 0<R≤∞, are exponentially close, that is, , 0<a<R, then q1≡q2 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.

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