Resolvent and spectrum for discrete symplectic systems in the limit point case

2021 ◽  
Author(s):  
Petr Zemánek
Keyword(s):  
Limit Point ◽  
Limit Point Case

A perturbation method and the limit-point case of even order symmetric differential expressions

1983 ◽  
Vol 94 (1-2) ◽  
pp. 121-135 ◽  
Author(s):  
Bernhard Mergler ◽  
Bernd Schultze
Keyword(s):  
Perturbation Method ◽  
Fourth Order ◽  
Limit Point ◽  
Perturbation Theorem ◽  
Order Case ◽  
Even Order ◽  
Limit Point Case ◽  
New Perturbation ◽  
Bounded Perturbations

SynopsisWe give a new perturbation theorem for symmetric differential expressions (relatively bounded perturbations, with relative bound 1) and prove with this theorem a new limit-point criterion generalizing earlier results of Schultze. We also obtain some new results in the fourth-order case.

Connection formulae for spectral functions associated with singular Dirac equations

2004 ◽  
Vol 134 (1) ◽  
pp. 215-223 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Dirac Equation ◽  
Spectral Function ◽  
Limit Point ◽  
Initial Condition ◽  
Spectral Functions ◽  
Dirac Equations ◽  
Special Cases ◽  
Limit Point Case ◽  
Image Position ◽  
Connection Formulae

We consider the Dirac equation given by with initial condition y1 (0) cos α + y2(0) sin α = 0, α ε [0; π ) and suppose the equation is in the limit-point case at infinity. Using to denote the derivative of the corresponding spectral function, a formula for is given when is known and positive for three distinct values of α. In general, if is known and positive for only two distinct values of α, then is shown to be one of two possibilities. However, in special cases of the Dirac equation, can be uniquely determined given for only two values of α.

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.

On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints

1989 ◽  
Vol 112 (3-4) ◽  
pp. 213-229 ◽  
Author(s):  
D. J. Gilbert
Keyword(s):  
Spectral Analysis ◽  
Asymptotic Behaviour ◽  
Limit Point ◽  
Absolutely Continuous ◽  
General Application ◽  
One Dimensional ◽  
Straightforward Method ◽  
Detailed Classification ◽  
Limit Point Case

SynopsisThe theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expressionL= –d2/(dr2) +V(r) is in the limit point case at both ends of an interval (a,b), withV(r) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions ofLu = xu, x εℝ, asr→aandr→b. The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.

A numerical method for the determination of the Titchmarsh-Weyl m -coefficient

1991 ◽  
Vol 435 (1895) ◽  
pp. 535-549 ◽  
Keyword(s):  
Numerical Method ◽  
Error Estimates ◽  
Complex Plane ◽  
Real Line ◽  
Integral Inequality ◽  
Limit Point ◽  
Computational Techniques ◽  
Limit Point Case ◽  
Null Set

A numerical method for determining the Titchmarsh-Weyl m ( λ ) function for the singular eigenvalue equation – ( py' )' + qy = λwy on [ a ,∞), where a is finite, is presented. The algorithm, based on Weyl’s theory, utilizes a result first used by Atkinson to map a point on the real line onto the Weyl circle in the complex plane. In the limit-point case these circles ‘nest’ and tend to the limit-point m ( λ ). Using Weyl’s result for the diameter of the circles, error estimates for m ( λ ) are obtained. In 1971, W. N. Everitt obtained an extension of an integral inequality of Hardy-Littlewood, namely the help inequality. He showed that the existence of that inequality is determined by the properties of the null set of Im[ λ 2 m ( λ )]. In view of the major difficulties in analysing m ( λ ) even in the rare cases when it is given explicitly, very few examples of the help inequality are known. The computational techniques discussed in this paper have been applied to the problem of finding best constants in these inequalities.

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.

Connection formulae for spectral functions associated with singular Sturm–Liouville equations

2000 ◽  
Vol 130 (1) ◽  
pp. 25-34 ◽  
Author(s):  
D. J. Gilbert ◽  
B. J. Harris
Keyword(s):  
Liouville Equation ◽  
Limit Point ◽  
Initial Condition ◽  
Spectral Functions ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Image Position ◽  
Almost All ◽  
Connection Formulae ◽  
Symmetric Derivative

We consider the Sturm–Liouville equation with the initial condition and suppose that Weyl's limit-point case holds at infinity. Let ρα(μ) be the corresponding spectral function and its symmetric derivative. We show that for almost all μ ∈ R, if exists and is positive for some α ∈ [0, π), then (i) exists and is positive for all β ∈ [0, π), and (ii) for all α1, α2 ∈ (0, π) \ {½ π},

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