On the solution of the inverse problem for a class of canonical systems corresponding to matrix string equations

We consider canonical systems (with 2p×2p Hamiltonians H(x) ≥ 0), which correspond to matrix string equations. Direct and inverse problems are solved in terms of Titchmarsh–Weyl and spectral matrix functions and related S-nodes. Procedures for solving inverse problems are given. MSC(2020): 34A55, 34B20, 34L40, 46N20, 47A48, 70H05

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

[2]  Vladimir S. Gerdjikov,et al.  Algebraic and Analytic Aspects of Soliton Type Equations , 2002, nlin/0206014.

[3]  V. Zakharov,et al.  Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II , 1979 .

[4]  Alexander L. Sakhnovich,et al.  New "Verblunsky-type" coefficients of block Toeplitz and Hankel matrices and of corresponding Dirac and canonical systems , 2019, J. Approx. Theory.

[5]  A. Constantin.  On the Inverse Spectral Problem for the Camassa–Holm Equation , 1998 .

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

[7]  C. Remling,et al.  Oscillation theory and semibounded canonical systems , 2018, Journal of Spectral Theory.

[8]  G. Teschl,et al.  Spectral asymptotics for canonical systems , 2014, 1412.0277.

[9]  A. Sakhnovich.  Dynamical canonical systems and their explicit solutions , 2016, 1603.08709.

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

[11]  C. Tretter,et al.  Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings , 2013, 1301.3408.

[12]  Order problem for canonical systems and a conjecture of Valent , 2015, 1502.04402.

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

[14]  Paul H. Rabinowitz,et al.  Periodic Solutions of Hamiltonian Systems: A Survey , 1982 .

[15]  H. Woracek,et al.  Strings, dual strings, and related canonical systems , 2007 .

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

[17]  V. Gerdjikov,et al.  Inverse scattering transform for the Camassa–Holm equation , 2006, Inverse Problems.

[18]  Alexander Sakhnovich.  Generalised canonical systems related to matrix string equations: corresponding structured operators and high-energy asymptotics of the Weyl functions , 2021 .

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

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

[21]  H. Langer.  Transfer Functions and Local Spectral Uniqueness for Sturm-Liouville Operators, Canonical Systems and Strings , 2016, 1604.00416.

[22]  L. Golinskii,et al.  Hilbert spaces of entire functions as a J theory subject , 1997 .

[23]  I. Kac.  The spectral property of dual matrix strings as a consequence of the existence of spectral matrix functions for canonical systems , 2000 .

[24]  Masatoshi Suzuki.  An inverse problem for a class of canonical systems having Hamiltonians of determinant one , 2020 .

[25]  H. Dym,et al.  Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations , 2012 .

[26]  Pseudospectral Functions for Canonical Differential Systems. II , 2012 .

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

[28]  A. Sakhnovich,et al.  Weyl functions of generalized dirac systems: Integral representation, the inverse problem and discrete interpolation , 2012 .

[29]  H. Langer,et al.  Direct and inverse spectral problems for generalized strings , 1998 .

[30]  Lev A. Sakhnovich,et al.  Factorization problems and operator identities , 1986 .

[31]  Jonathan Eckhardt,et al.  The inverse spectral problem for indefinite strings , 2014, 1409.0139.

[32]  I. S. Kats.  SPECTRAL FUNCTIONS OF A STRING , 1983 .

[33]  Harald Woracek,et al.  Canonical systems with discrete spectrum , 2019, 1904.03662.

[34]  Jonathan Eckhardt,et al.  The Classical Moment Problem and Generalized Indefinite Strings , 2017, 1707.08394.

[35]  H. Langer,et al.  Continuation of Hermitian Positive Definite Functions and Related Questions , 2014 .