scholarly journals J-Self-Adjoint Extensions for a Class of Discrete Linear Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Guojing Ren ◽  
Huaqing Sun
Keyword(s):  
Boundary Conditions ◽  
Hamiltonian Systems ◽  
Limit Point ◽  
Limit Circle ◽  
Linear Hamiltonian Systems ◽  
Infinite Intervals ◽  
Defect Indices

This paper is concerned with formallyJ-self-adjoint discrete linear Hamiltonian systems on finite or infinite intervals. The minimal and maximal subspaces are characterized, and the defect indices of the minimal subspaces are discussed. All theJ-self-adjoint subspace extensions of the minimal subspace are completely characterized in terms of the square summable solutions and boundary conditions. As a consequence, characterizations of all theJ-self-adjoint subspace extensions are given in the limit point and limit circle cases.

The HELP inequality for lim-p Hamiltonian systems

2000 ◽  
Vol 130 (4) ◽  
pp. 699-720 ◽  
Author(s):  
B. M. Brown ◽  
M. Marletta
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hamiltonian Systems ◽  
Wide Class ◽  
Limit Point ◽  
Type Singularity ◽  
Linear Hamiltonian Systems ◽  
Infinite Intervals ◽  
Point Type

In a recent paper, Brown, Evans and Marletta extended the HardyEverittLittlewoodPolya inequality from 2nth-order formally self-adjoint ordinary differential equations to a wide class of linear Hamiltonian systems in 2n variables. The paper considered only problems on semi-infinite intervals [a, ∞) with a limit-point type singularity at infinity. In this paper we extend the theory to cover all types of endpoint ( lim-p for n ≤ p ≤ 2n ).

scholarly journals Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yan Liu ◽  
Meiru Xu
Keyword(s):  
Perturbation Theory ◽  
Hamiltonian Systems ◽  
Limit Point ◽  
Limit Circle ◽  
Linear Relations ◽  
Deficiency Indices ◽  
Bounded Perturbations

AbstractThis paper is concerned with stability of deficiency indices for discrete Hamiltonian systems under perturbations. By applying the perturbation theory of Hermitian linear relations we establish the invariance of deficiency indices for discrete Hamiltonian systems under bounded perturbations. As a consequence, we obtain the invariance of limit types for the systems under bounded perturbations. In particular, we build several criteria of the invariance of the limit circle and limit point cases for the systems. Some of these results improve and extend some previous results.

scholarly journals The Oscillation Numbers and the Abramov Method of Spectral Counting for Linear Hamiltonian Systems

2021 ◽  
Vol 248 ◽  
pp. 01002
Author(s):  
Julia Elyseeva
Keyword(s):  
Boundary Conditions ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Spectral Counting ◽  
Hamiltonian Problems ◽  
Differential Systems ◽  
Linear Hamiltonian Systems ◽  
Strict Normality

In this paper we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. For the Hamiltonian problems we do not assume any controllability and strict normality assumptions which guarantee that the classical eigenvalues of the problems are isolated. We also omit the Legendre condition for their Hamiltonians. We show that the Abramov method of spectral counting can be modified for the more general case of finite eigenvalues of the Hamiltonian problems and then the constructive ideas of the Abramov method can be used for stable calculations of the oscillation numbers and finite eigenvalues of the Hamiltonian problems.

Parameterization of the M(λ) function for a Hamiltonian system of limit circle type

1983 ◽  
Vol 93 (3-4) ◽  
pp. 349-360 ◽  
Author(s):  
D. B. Hinton ◽  
J. K. Shaw
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Deficiency Index ◽  
Limit Circle ◽  
Weyl Matrix ◽  
Linear Hamiltonian Systems ◽  
Limit Circle Case ◽  
Special Case ◽  
Circle Case

SynopsisThe authors continue their study of Titchmarch-Weyl matrix M(λ) functions for linear Hamiltonian systems. A representation for the M(λ) function is obtained in the case where the system is limit circle, or maximum deficiency index, type. The representation reduces, in a special case, to a parameterization for scalar m-coefficients due to C. T. Fulton. A proof that matrix M(λ) functions are meromorphic in the limit circle case is given.

scholarly journals Extensions of symmetric singular second-order dynamic operators on time scales

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1475-1484 ◽  
Author(s):  
Bilender Allakhverdiev
Keyword(s):  
Boundary Conditions ◽  
Time Scales ◽  
Limit Point ◽  
Second Order ◽  
Boundary Values ◽  
Infinite Time ◽  
Limit Circle ◽  
Symmetric Operators

A space of boundary values is constructed for minimal symmetric singular second-order dynamic operators on semi-infinite and infinite time scales in limit-point and limit-circle cases. A description of all maximal dissipative, maximal accumulative, selfadjoint, and other extensions of such symmetric operators is given in terms of boundary conditions.

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