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On local Borg-Marchenko uniqueness results. (English) Zbl 0985.34077

The authors give a new proof of the fact that if the Weyl-Titchmarsh functions \(m_1\) and \(m_2\) of two 1D Schrödinger operators \(-d^2/dx^2+q_j(x)\), \(j=1\), 2, in \(L^2([0,\infty))\) with Dirichlet condition at \(x=0\) and potentials integrable on \((0,R)\) for all \(R>0\), satisfy \(|m_1(z)-m_2(z)|=O(\exp\{-2a\text{Im} \sqrt z\})\) for some \(a>0\) as \(|z|\to\infty\) along a complex ray, then \(q_1=q_2\) a.e.on \([0,a]\). Variants for bounded intervals, more general boundary conditions, and for systems are included.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L25 Scattering theory, inverse scattering involving ordinary differential operators
34A55 Inverse problems involving ordinary differential equations
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