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, m_j(z), of two Schrödinger operators, H_j = -d^2/dx^2 + q_j, j = 1,2 in L^2((O,R)), O < R ≤ ∞, are exponentially close, that is, |m_1(z) - m_2(z)|_|z|→∞ = O(e^(-2 IM(z^1/2)a), O < ɑ <R, then q_1 = q_2 a.e. on [O,ɑ]. The result applies to any boundary conditions at x = O 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.