On a Weyl-Titchmarsh theory for discrete symplectic systems on a half line

Recently, Bohner and Sun introduced basic elements of a Weyl-Titchmarsh theory into the study of discrete symplectic systems. We extend this development through the introduction of Weyl-Titchmarsh functions together with a preliminary study of their properties. A limit point criterion is described and characterized. Green's function for the half-line is introduced as a limit of such functions in the regular case and half-line solutions obtained are seen to satisfy lambda-dependent boundary conditions at infinity.

[1]  V. I. Kogan,et al.  1.—On Square-integrable Solutions of Symmetric Systems of Differential Equations of Arbitrary Order. , 1976, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[2]  Steve Clark,et al.  Weyl-Titchmarsh M-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators , 2001 .

[3]  A. G. Kostyuchenko,et al.  Generalized Jacobi matrices and deficiency numbers of ordinary differential operators with polynomial coefficients , 1999 .

[4]  Allan M. Krall,et al.  A limit-point criterion for linear hamiltonian systems , 1996 .

[5]  A. S. Osipov.  Some Properties of Resolvent Sets of Second-Order Difference Operators with Matrix Coefficients , 2000 .

[6]  Walter Van Assche,et al.  Orthogonal matrix polynomials and higher-order recurrence relations , 1993 .

[7]  Martin Bohner,et al.  Weyl-Titchmarsh theory for symplectic difference systems , 2010, Appl. Math. Comput..

[8]  J. K. Shaw,et al.  Hamiltonian systems of limit point or limit circle type with both endpoints singular , 1983 .

[9]  F. V. Atkinson,et al.  Discrete and Continuous Boundary Problems , 1964 .

[10]  Albert Schneider,et al.  On the Titchmarsh‐Weyl Coefficients for Singular S‐Hermitian Systems II , 1993 .

[11]  Helge Holden,et al.  Borg-Type Theorems for Matrix-Valued Schrödinger Operators , 1999 .

[12]  Werner Kratz,et al.  Quadratic Functionals in Variational Analysis and Control Theory , 1995 .

[13]  Rafael Obaya,et al.  Ergodic properties and Weyl M-functions for random linear Hamiltonian systems , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[14]  L. Markus,et al.  Complex symplectic geometry with applications to ordinary differential operators , 1999 .

[15]  Martin Bohner,et al.  Disconjugacy and Transformations for Symplectic Systems , 1997 .

[16]  Vera Zeidan,et al.  Nonnegativity of discrete quadratic functionals corresponding to symplectic difference systems , 2003 .

[17]  Stephen Clark,et al.  Borg-Marchenko-type Uniqueness Results for CMV Operators , 2008, 0803.3175.

[18]  Vera Zeidan,et al.  Extension of Discrete LQR-Problem to Symplectic Systems ∗ , 2007 .

[19]  Steve Clark,et al.  On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems , 2003, math/0312177.

[20]  J. K. Shaw,et al.  On Boundary Value Problems for Hamiltonian Systems with Two Singular Points , 1984 .

[21]  J. K. Shaw,et al.  ON THE SPECTRUM OF A SINGULAR HAMILTONIAN SYSTEM , 1982 .

[22]  A. G. Kostyuchenko,et al.  Complete Indefiniteness Tests for Jacobi Matrices with Matrix Entries , 2001 .

[23]  Antonio J. Durán,et al.  ORTHOGONAL MATRIX POLYNOMIALS: ZEROS AND BLUMENTHAL'S THEOREM , 1996 .

[24]  J. K. Shaw,et al.  Parameterization of the M(λ) function for a Hamiltonian system of limit circle type , 1983, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[25]  Martin Bohner,et al.  Sturmian and spectral theory for discrete symplectic systems , 2009 .

[26]  Martin Bohner,et al.  An Oscillation Theorem for Discrete Eigenvalue Problems , 2003 .

[27]  A. Aptekarev,et al.  The Scattering Problem for a Discrete Sturm-Liouville Operator , 1984 .

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

[29]  Roman Šimon Hilscher,et al.  Riccati inequality and other results for discrete symplectic systems , 2006 .

[30]  A. G. Kostyuchenko,et al.  Three-term recurrence relations with matrix coefficients. The completely indefinite case , 1998 .

[31]  Steve Clark,et al.  Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators , 2004, math/0408074.

[32]  A. Sakhnovich,et al.  SPECTRAL FUNCTIONS OF A CANONICAL SYSTEM OF ORDER $ 2n$ , 1992 .

[33]  Fritz Gesztesy,et al.  An Addendum to Krein's Formula , 1997 .

[34]  H. Weyl,et al.  Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen , 1910 .

[35]  Werner Kratz,et al.  Oscillation and spectral theory for symplectic difference systems with separated boundary conditions , 2010 .

[36]  E. Hille,et al.  Lectures on ordinary differential equations , 1968 .

[37]  S. A. Orlov,et al.  NESTED MATRIX DISKS ANALYTICALLY DEPENDING PARAMETER, AND THEOREMS ON THE INVARIANCE RADII OF LIMITING DISKS , 1976 .

[38]  Stephen Clark,et al.  WEYL-TITCHMARSH THEORY AND BORG-MARCHENKO-TYPE UNIQUENESS RESULTS FOR CMV OPERATORS WITH MATRIX-VALUED , 2010, 1002.0387.

[39]  H. Dym,et al.  Operator theory: Advances and applications , 1991 .

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

[41]  Matthias Lesch,et al.  On the deficiency indices and self-adjointness of symmetric Hamiltonian systems , 2003 .

[42]  Vera Zeidan,et al.  Symplectic difference systems: variable stepsize discretization and discrete quadratic functionals , 2003 .

[43]  Lawrence Markus,et al.  Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-differential Operators , 1998 .

[44]  Jeffrey S. Geronimo,et al.  Scattering theory and matrix orthogonal polynomials on the real line , 1982 .

[45]  Yuming Shi,et al.  Weyl–Titchmarsh theory for a class of discrete linear Hamiltonian systems , 2006 .

[46]  Steve Clark,et al.  Weyl-titchmarsh M -function Asymptotics for Matrix-valued Schr¨odinger Operators , 2022 .

[47]  J. Weidmann,et al.  Spectral Theory of Ordinary Differential Operators , 1987 .

[48]  Antonio J. Durán,et al.  N-EXTREMAL MATRICES OF MEASURES FOR AN INDETERMINATE MATRIX MOMENT PROBLEM , 2000 .

[49]  Israel Gohberg,et al.  Inverse spectral problems for difference operators with rational scattering matrix function , 1994 .

[50]  P. López-Rodríguez.  Riesz's Theorem for Orthogonal Matrix Polynomials , 1999 .

[51]  Andrei Osipov.  On some issues related to the moment problem for the band matrices with operator elements , 2002 .

[52]  J. K. Shaw,et al.  On Titchmarsh-Weyl M(λ)-functions for linear Hamiltonian systems , 1981 .

[53]  Allan M. Krall,et al.  M (λ) theory for singular Hamiltonian systems with two singular points , 1989 .

[54]  Werner Kratz,et al.  Oscillation theorems for symplectic difference systems , 2007 .

[55]  Masahisa Fukushima,et al.  A spectral representation on ordinary linear difference equation with operator‐valued coefficients of the second order , 1976 .

[56]  R. Bolstein,et al.  Expansions in eigenfunctions of selfadjoint operators , 1968 .

[57]  C. Ahlbrandt,et al.  Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations , 1996 .

[58]  Vera Zeidan,et al.  Coupled intervals for discrete symplectic systems , 2006 .