Article
in "Spectral Theory and Differential Equations: V.A. Marchenko 90th Anniversary Collection", E. Khruslov, L. Pastur, and D. Shepelsky (eds), 117-133, Advances in the Mathematical Sciences 233, Amer. Math. Soc., Providence, 2014. [url]

Inverse uniqueness results for one-dimensional weighted Dirac operators

Jonathan Eckhardt, Aleksey Kostenko, and Gerald Teschl

Abstract
Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl-Titchmarsh-Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one regular endpoint. Moreover, our result also improves the currently known results for canonical (Hamiltonian) systems. If one endpoint is in the limit circle case, we also establish corresponding two-spectra results.

MSC2010: Primary 34L40, 34B20; Secondary 46E22, 34A55
Keywords: Dirac operators, canonical systems, inverse spectral theory, de Branges spaces

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