Hamiltonian systems of limit point or limit circle type with both endpoints singular

Abstract A linear Hamiltonian system J y′ = ( λA + B ) y is considered on an open interval ( a , b ), where both a and b are singular. The system is assumed to be of limit point or limit circle type at the endpoints. A theory of boundary problems for such systems is developed. Explicit boundary conditions are given, resolvent operators constructed and unique solutions established. The results given extend to Hamiltonian systems a theory of singular boundary value problems due to M. H. Stone and K. Kodaira.

[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]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

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

[4]  K. Kodaira ( 18 ) The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices , 1949 .

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

[6]  Charles T. Fulton,et al.  Parametrizations of Titchmarsh’s ()-functions in the limit circle case , 1977 .

[7]  D. Hinton,et al.  Titchmarsh-Weyl Theory for Hamiltonian Systems , 1981 .

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

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

[10]  ADJOINT BOUNDARY VALUE PROBLEMS FOR COMPACTIFIED SINGULAR DIFFERENTIAL OPERATORS , 1973 .

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

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

[13]  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.

[14]  Boundary values for an eigenvalue problem with a singular potential , 1982 .

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

[16]  M. Stone Linear transformations in Hilbert space and their applications to analysis , 1932 .