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.

Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators

2016 ◽  
Vol 66 (4) ◽  
Author(s):  
Bilender P. Allahverdiev
Keyword(s):  
Difference Operators ◽  
Boundary Values ◽  
Defect Index ◽  
Order Difference ◽  
Limit Circle ◽  
Limit Circle Case ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Circle Case ◽  
Liouville Operators

AbstractIn this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index (1, 1) (in limit-circle case at ±∞ and limit-point case at ∓∞), acting in the Hilbert space

m (λ)-functions for complex Sturm-Liouville operators

1980 ◽  
Vol 86 (3-4) ◽  
pp. 275-289 ◽  
Author(s):  
David Race
Keyword(s):  
Limit Point ◽  
Limit Circle ◽  
Limit Circle Case ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Complex Valued ◽  
Well Posed ◽  
Circle Case ◽  
Liouville Operators

SynopsisIn this paper, the formally J-symmetric Sturm-Liouville operator with complex-valued coefficients is considered and a generalisation of the Weyl limit-point, limit-circle dichotomy is sought by means of m (λ )-functions. These functions are then used to give an explicit description of all the associated J-selfadjoint operators with separated boundary conditions in the limit-circle case. A formulation of the eigenvalues of these operators, and a characterisation of which extensions are non-well-posed, are also found. Finally, the limit-point case is studied, mainly by means of an example.

HELP inequalities for limit-circle and regular problems

1991 ◽  
Vol 432 (1886) ◽  
pp. 367-390 ◽  
Keyword(s):  
Differential Equation ◽  
Limit Point ◽  
Strong Limit ◽  
End Points ◽  
Limit Circle ◽  
End Point ◽  
Limit Circle Case ◽  
Limit Point Case ◽  
Regular Problems ◽  
Circle Case

Everitt’s criterion for the validity of the generalized Hardy-Littlewood inequality presupposes that the associated differential equation is singular at one end-point of the interval of definition and is in the strong-limit-point case at the end-point. In this paper we investigate the cases when the differential equation is in the limit-circle case and non-oscillatory at the singular end-point and when both end-points of the interval are regular.

scholarly journals Weyl-Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

2011 ◽  
Vol 2011 ◽  
pp. 1-41 ◽  
Author(s):  
Roman Šimon Hilscher ◽  
Petr Zemánek
Keyword(s):  
Limit Point ◽  
Time Scale ◽  
Original System ◽  
Limit Circle ◽  
Limit Circle Case ◽  
Nonhomogeneous System ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
The Green Function ◽  
Circle Case

We develop the Weyl-Titchmarsh theory for time scale symplectic systems. We introduce theM(λ)-function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl-Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second-order Sturm-Liouville equations on time scales.

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.

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