Dirac systems with locally square-integrable potentials: direct and inverse problems for the spectral functions

We solve the inverse problems to recover Dirac systems on an interval or semiaxis from their spectral functions (matrix valued functions) for the case of locally square-integrable potentials. Direct problems in terms of spectral functions are treated as well. Moreover, we present necessary and sufficient conditions on the given distribution matrix valued function to be a spectral function of some Dirac system with a locally square-integrable potential. Interesting connections with Paley-Wiener sampling measures appear in the case of scalar spectral functions.

[1]  N. Makarov,et al.  Etudes for the inverse spectral problem , 2022, Journal of the London Mathematical Society.

[2]  Lev Sakhnovich,et al.  Characteristic function of M. Liv\v{s}ic and some developments , 2021, 2104.12694.

[3]  Masatoshi Suzuki.  An inverse problem for a class of lacunary canonical systems with diagonal Hamiltonian , 2019, Tohoku Mathematical Journal.

[4]  F. Gesztesy,et al.  The inverse approach to Dirac-type systems based on the A-function concept , 2019, Journal of Functional Analysis.

[5]  A. Sakhnovich.  Scattering for general-type Dirac systems on the semi-axis: reflection coefficients and Weyl functions , 2018, Journal of Differential Equations.

[6]  L. Sakhnovich,et al.  Dirac Equation: the Stationary and Dynamical Scattering Problems , 2016, 1611.03458.

[7]  F. Gesztesy,et al.  Renormalized oscillation theory for Hamiltonian systems , 2016, 1608.02116.

[8]  G. Teschl,et al.  On Spectral Deformations and Singular Weyl Functions for One-Dimensional Dirac Operators , 2014, 1410.1152.

[9]  G. Teschl,et al.  Inverse spectral problems for Schrödinger-type operators with distributional matrix-valued potentials , 2014, Differential and Integral Equations.

[10]  A. Sakhnovich.  Inverse problem for Dirac systems with locally square-summable potentials and rectangular Weyl functions , 2014, 1401.3605.

[11]  L. Sakhnovich,et al.  Inverse Problems and Nonlinear Evolution Equations: Solutions, Darboux Matrices and Weyl-Titchmarsh Functions , 2013 .

[12]  A. Sakhnovich,et al.  Recovery of the Dirac system from the rectangular Weyl matrix function , 2011, 1106.1263.

[13]  H. Woracek,et al.  A local inverse spectral theorem for Hamiltonian systems , 2011 .

[14]  Y. Mykytyuk,et al.  Inverse spectral problems for Dirac operators on a finite interval , 2011, 1101.2302.

[15]  R. Hryniv,et al.  Reconstruction of radial Dirac operators , 2007 .

[16]  F. Gesztesy,et al.  On self-adjoint and J-self-adjoint Dirac-type operators: a case study , 2005, math/0511369.

[17]  Israel Gohberg,et al.  Theory and Applications of Volterra Operators in Hilbert Space , 2004 .

[18]  Alexander Sakhnovich,et al.  Dirac type and canonical systems: spectral and Weyl–Titchmarsh matrix functions, direct and inverse problems , 2002 .

[19]  K. Seip,et al.  On Fourier frames , 2000, math/0005092.

[20]  Lev A. Sakhnovich,et al.  Spectral Theory of Canonical Differential Systems. Method of Operator Identities , 1999 .

[21]  Lev A. Sakhnovich,et al.  Interpolation Theory and Its Applications , 1997 .

[22]  Anthony Leonard,et al.  INTEGRAL EQUATIONS WITH DIFFERENCE KERNELS ON FINITE INTERVALS. , 1965 .

[23]  Matthias Abend,et al.  Volterra Integral And Functional Equations , 2016 .

[24]  Louis de Branges,et al.  Hilbert spaces of entire functions , 1968 .

[25]  H. Holden,et al.  Institute for Mathematical Physics Abstract Wave Equations and Associated Dirac-type Operators Abstract Wave Equations and Associated Dirac-type Operators , 2022 .