scholarly journals Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Kun Li ◽  
Maozhu Zhang ◽  
Jinming Cai ◽  
Zhaowen Zheng
Keyword(s):  
Boundary Condition ◽  
Dirac Operator ◽  
Completeness Theorem ◽  
Scattering Matrix ◽  
Dissipative Operator ◽  
Limit Circle ◽  
Associated Functions ◽  
Limit Circle Case ◽  
Eigenfunctions And Associated Functions ◽  
Circle Case

This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.

The limit-circle case for a positive definite $J$ -fraction

1945 ◽  
Vol 12 (2) ◽  
pp. 255-273 ◽  
Author(s):  
Joseph J. Dennis ◽  
H. S. Wall
Keyword(s):  
Positive Definite ◽  
Limit Circle ◽  
Limit Circle Case ◽  
Circle Case

scholarly journals Second-Order Differential Operators in the Limit Circle Case

2021 ◽  
Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Measures ◽  
Limit Circle ◽  
Jacobi Operators ◽  
Limit Circle Case ◽  
Circle Case ◽  
Stieltjes Transforms

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

The spectra of well-posed operators

1995 ◽  
Vol 125 (6) ◽  
pp. 1331-1348 ◽  
Author(s):  
Sobhy El-sayed Ibrahim
Keyword(s):  
Adjoint Equation ◽  
Integral Operators ◽  
Essential Spectra ◽  
Limit Circle ◽  
All Solutions ◽  
Limit Circle Case ◽  
Complex Coefficients ◽  
Well Posed ◽  
Circle Case ◽  
Quasidifferential Expression

In this paper, the general ordinary quasidifferential expression M of nth order, with complex coefficients, and its formal adjoint M− are considered. It is shown in the case of two singular endpomts and when all solutions of the equation and the adjoint equation are in (the limit-circle case) that all well-posed extensions of the minimal operator T0(M) have resolvents which are Hilbert Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all of the standard essential spectra to be empty. These results extend those for the formally symmetric expression M studied in [1] and [14], and also extend those proved in [8] for one singular endpoint.

scholarly journals An Alternate Proof of the De Branges Theorem on Canonical Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Keshav Raj Acharya
Keyword(s):  
Canonical System ◽  
Alternative Proof ◽  
Defect Index ◽  
Alternate Proof ◽  
Symmetric Relation ◽  
Canonical Systems ◽  
Limit Circle ◽  
Limit Circle Case ◽  
Limit Point Case ◽  
Circle Case

The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on ℂ. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.

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.

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