Inverse uniqueness results for one-dimensional weighted Dirac operators

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 twospectra results.2010 Mathematics Subject Classification. Primary 34L40, 34B20; Secondary 46E22, 34A55.

Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators

“…We remark that similar results can be proven using the theory of de Branges spaces [9,14]. However, our assumptions on Φ(z, x) here are of a different nature and more convenient to derive in certain situations; [16,Section 6].…”

“…An alternate approach using the theory of de Branges spaces will be given in [14] and spectral asymptotics for the singular Weyl functions will be given in [15].…”

Spectral asymptotics for canonical systems

“…To complete this subsection, let us consider the spectral problem (2.1) also without the trace normalization assumption (2.10). The next result is an analog of Theorem 2.2 (see, e.g., [17,60]) in this case.…”

On spectral deformations and singular Weyl functions for one-dimensional Dirac operators

“…It has recently proven to be a powerful tool for inverse spectral theory for these operators and further refinements were given in [3], [4], [5], [6], [7], [10], [20]. The analogous theory for one-dimensional Dirac operators was developed by Brunnhuber and three of us in [2] (for further extensions see also [8], [9]). Nevertheless, such operators are still difficult to understand.…”