scholarly journals Dissipative Operators Generated by the Sturm–Liouville Differential Expression in the Weyl Limit Circle Case

2001 ◽  
Vol 254 (1) ◽  
pp. 178-190 ◽  
Author(s):  
Elgiz Bairamov ◽  
Allan M. Krall
Keyword(s):  
Differential Expression ◽  
Limit Circle ◽  
Dissipative Operators ◽  
Limit Circle Case ◽  
Sturm Liouville ◽  
Circle Case

NONLINEAR SINGULAR STURM-LIOUVILLE PROBLEMS WITH IMPULSIVE CONDITIONS

2019 ◽  
pp. 439
Author(s):  
Bilender P. Allahverdiev ◽  
Husein Tuna
Keyword(s):  
Existence And Uniqueness ◽  
Liouville Problem ◽  
Uniqueness Theorems ◽  
Impulsive Conditions ◽  
Limit Circle ◽  
Existence And Uniqueness Theorems ◽  
Non Linear ◽  
Limit Circle Case ◽  
Sturm Liouville ◽  
Circle Case

In this paper, we consider a non-linear impulsive Sturm-Liouville problem on semiinfinite intervals in which the limit-circle case holds at infinity for THE Sturm-Liouville expression. We prove the existence and uniqueness theorems for this problem.

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.

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.

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

Livšic’s theorem for q-Sturm—Liouville operators

2016 ◽  
Vol 53 (4) ◽  
pp. 512-524
Author(s):  
Hüseyin Tuna ◽  
Aytekin Eryilmaz
Keyword(s):  
Limit Circle ◽  
Limit Circle Case ◽  
Sturm Liouville ◽  
Circle Case ◽  
Liouville Operators

In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

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.

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