Overview of Weyl-Titchmarsh Theory for Second Order Sturm-Liouville Equations on Time Scales

In this paper we present an overview of the basic Weyl-Titchmarsh theory for second order Sturm-Liouville equations on time scales. We construct m(lambda)-function, the Weyl solution, and Weyl disk. We justify the terminology ``disk'' by its geometric properties, show explicitly the coordinates of the center of the disk, and calculate its radius. We show that the dichotomy regarding the square-integrable solutions known in the continuous time and discrete theory works in the same way for general time scales.

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

[2]  Yoshimi Saito,et al.  Eigenfunction Expansions Associated with Second-order Differential Equations for Hilbert Space-valued Functions , 1971 .

[3]  J. Weiss,et al.  Limit-Point Criteria for a Second Order Dynamic Equation on Time Scales , 2009 .

[4]  Yuming Shi.  On the rank of the matrix radius of the limiting set for a singular linear Hamiltonian system , 2004 .

[5]  Yuming Shi,et al.  Limit-point and limit-circle criteria for singular second-order linear difference equations with complex coefficients , 2006, Comput. Math. Appl..

[6]  Petr Zemánek,et al.  Weyl-Titchmarsh Theory for Time Scale Symplectic Systems on Half Line , 2011 .

[7]  E. C. Titchmarsh.  ON EXPANSIONS IN EIGENFUNCTIONS (II) , 1941 .

[8]  Jiangang Qi,et al.  Strong limit-point classification of singular Hamiltonian expressions , 2004 .

[9]  Vera Zeidan,et al.  Calculus of variations on time scales: weak local piecewise Crd1 solutions with variable endpoints , 2004 .

[10]  Shejie Lu,et al.  Classification for a class of second-order singular equations on time scales , 2007 .

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

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

[13]  T. Asahi.  Spectral Theory of the Difference Equations , 1966 .

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

[15]  Stephen Clark,et al.  On a Weyl-Titchmarsh theory for discrete symplectic systems on a half line , 2010, Appl. Math. Comput..

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

[17]  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.

[18]  S. Clark,et al.  A Spectral Analysis for Self-Adjoint Operators Generated by a Class of Second Order Difference Equations , 1996 .

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

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

[21]  Allen Sims,et al.  Secondary Conditions for Linear Differential Operators of the Second Order , 1957 .

[22]  Orthogonal polynomials and extensions of Copson's inequality , 1993 .

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

[24]  J. K. Shaw,et al.  The Asymptotic Form of the Titchmarsh‐Weyl Coefficient for Dirac Systems , 1983 .

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

[26]  Adil Huseynov,et al.  LIMIT POINT AND LIMIT CIRCLE CASES FOR DYNAMIC EQUATIONS ON TIME SCALES , 2010 .

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

[28]  Philip W. Walker.  A Vector-Matrix Formulation for Formally Symmetric Ordinary Differential Equations with Applications to Solutions of Integrable Square , 1974 .

[29]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[30]  Martin Bohner,et al.  Weyl-Titchmarsh Theory for Hamiltonian Dynamic Systems , 2010 .

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

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

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

[34]  Chao Zhang,et al.  Classification for a class of second-order singular equations on time scales , 2007, Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing (SNPD 2007).

[35]  Yuming Shi,et al.  Strong limit point criteria for a class of singular discrete linear Hamiltonian systems , 2007 .

[36]  William Norrie Everitt,et al.  A personal history of the m -coefficient , 2004 .

[37]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

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

[39]  Michael Plum,et al.  Titchmarsh–Sims–Weyl Theory for Complex Hamiltonian Systems , 2003 .

[40]  W. D. Evans,et al.  On an extension of Copson's inequality for infinite series , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[41]  G. Weiss,et al.  EIGENFUNCTION EXPANSIONS. Associated with Second-order Differential Equations. Part I. , 1962 .

[42]  E. C. Titchmarsh,et al.  Reviews , 1947, The Mathematical Gazette.

[43]  A. Peterson,et al.  Advances in Dynamic Equations on Time Scales , 2012 .

[44]  C. Remling.  Geometric Characterization of Singular Self-Adjoint Boundary Conditions for Hamiltonian Systems , 1996 .

[45]  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.