Inverse uniqueness results for Schr\"odinger operators using de Branges
  theory

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Abstract

We utilize the theory of de Branges spaces to show when certain Schr\"odinger operators with strongly singular potentials are uniquely determined by their associated spectral measure. The results are applied to obtain an inverse uniqueness theorem for perturbed spherical Schr\"odinger operators.

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On spectral theory for Schrödinger operators with strongly singular potentials

Fritz Gesztesy, Maxim Zinchenko (2006)

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On the Singular Weyl-Titchmarsh Function of Perturbed Spherical Schroedinger Operators

Gerald Teschl, Aleksey Kostenko (2010)

We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schroedinger operators (also known as Bessel operators) under the assumption that the perturbation \(q(x)\) satisfies \(x q(x) \in L^1(0,1)\). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.