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

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.

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.

scholarly journals q-Titchmarsh-Weyl theory: series expansion

2012 ◽  
Vol 205 ◽  
pp. 67-118
Author(s):  
M. H. Annaby ◽  
Z. S. Mansour ◽  
I. A. Soliman
Keyword(s):  
Limit Point ◽  
Eigenfunction Expansion ◽  
Bessel Functions ◽  
Sufficient Conditions ◽  
Limit Circle ◽  
Worked Example ◽  
Weyl Theory ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Theory Series

AbstractWe establish aq-Titchmarsh-Weyl theory for singularq-Sturm-Liouville problems. We defineq-limit-point andq-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jacksonq-Bessel functions is given. This example leads to the completeness of a wide class ofq-cylindrical functions.

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 q-Titchmarsh-Weyl theory: series expansion

2012 ◽  
Vol 205 ◽  
pp. 67-118 ◽  
Author(s):  
M. H. Annaby ◽  
Z. S. Mansour ◽  
I. A. Soliman
Keyword(s):  
Limit Point ◽  
Eigenfunction Expansion ◽  
Bessel Functions ◽  
Sufficient Conditions ◽  
Limit Circle ◽  
Worked Example ◽  
Weyl Theory ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Theory Series

AbstractWe establish a q-Titchmarsh-Weyl theory for singular q-Sturm-Liouville problems. We define q-limit-point and q-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jackson q-Bessel functions is given. This example leads to the completeness of a wide class of q-cylindrical functions.

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