Limit-point and limit-circle criteria for Sturm-Liouville equations with intermittently negative principal coefficients

1986 ◽  
Vol 103 (3-4) ◽  
pp. 215-228 ◽  
Author(s):  
W. N. Everitt ◽  
I. W. Knowles ◽  
T. T. Read
Keyword(s):  
Differential Expression ◽  
Limit Point ◽  
Limit Circle ◽  
Almost Everywhere ◽  
Sturm Liouville ◽  
Image Position

SynopsisLimit-point and limit-circle criteria are given for the generalised Sturm-Liouville differential expressionwhere(i) p, q, and w are real-valued on [a, b),(ii) p−1, q, w are locally Lebesgue integrable on [a, b),(iii) w > 0 almost everywhere on [a, b) and the principal coefficient p is allowed toassume both positive and negative values.

17.—The Limit-point and Limit-circle Cases for Polynomials in a Differential Operator

1975 ◽  
Vol 72 (3) ◽  
pp. 219-224 ◽  
Author(s):  
Anton Zettl
Keyword(s):  
Differential Operator ◽  
Differential Expression ◽  
Limit Point ◽  
Half Line ◽  
Limit Circle

SynopsisThis paper is concerned with the L2 classification of ordinary symmetrical differential expressions defined on a half-line [0, ∞) and obtained from taking formal polynomials of symmetric differential expression. The work generalises results in this area previously obtained by Chaudhuri, Everitt, Giertz and the author.

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.

Dissipative Sturm-Liouville operators

1981 ◽  
Vol 88 (3-4) ◽  
pp. 329-343 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Minimal Operator ◽  
Sturm Liouville ◽  
Image Position ◽  
Complex Valued ◽  
Liouville Operators

SynopsisConsider the differential expressionwherepandw> 0 are real-valued andqis complex-valued onI. A number of criteria are established for certain extensions of the minimal operator generated by τ in the weighted Hilbert spaceto be maximal dissipative.

Limit Point and Limit Circle Criteria for a Class of Singular Symmetric Differential Operators

1976 ◽  
Vol 28 (5) ◽  
pp. 905-914 ◽  
Author(s):  
Robert L. Anderson
Keyword(s):  
Limit Point ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Limit Circle ◽  
Image Position ◽  
Point Type ◽  
Symmetric Differential Operators

For certain classes of singular symmetric differential operators L of order 2n, this paper considers the problem of determining sufficient conditions for L to be of limit point type or of limit circle type. The operator discussed here is defined by

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},

1.—An Algorithm for the Construction of Limit Circle Expressions

1976 ◽  
Vol 75 (1) ◽  
pp. 1-3
Author(s):  
Anton Zettl
Keyword(s):  
Differential Expression ◽  
Limit Circle ◽  
All Solutions ◽  
Image Position ◽  
Parameter Function

SynopsisConsider the differential expressionIt is shown here that given pn > 0 with pn ∈ C2n(0, ∞) there exist coefficientssuch that all solutions of the equation Ry = 0 are in L2(0,∞). The pi for i<n can be explicitly obtained in terms of pn and a parameter function.

Weighted Estimates for Solutions of the General Sturm-Liouville Equation and the Everitt-Giertz Problem. I

2014 ◽  
Vol 58 (1) ◽  
pp. 125-147 ◽  
Author(s):  
N. A. Chernyavskaya ◽  
L. A. Shuster
Keyword(s):  
Liouville Equation ◽  
Positive Constant ◽  
Absolute Constant ◽  
Absolutely Continuous ◽  
Weighted Estimates ◽  
Negative Function ◽  
Almost Everywhere ◽  
Absolute Positive Constant ◽  
Sturm Liouville ◽  
Image Position

AbstractConsider the equationwhereƒ∈Lp(ℝ),p∈ (1, ∞) andBy a solution of (*), we mean any functionyabsolutely continuous together with (ry′) and satisfying (*) almost everywhere on ℝ. In addition, we assume that (*) is correctly solvable in the spaceLp(ℝ), i.e.(1) for any function, there exists a unique solutiony∈Lp(ℝ) of (*);(2) there exists an absolute constantc1(p) > 0 such that the solutiony∈Lp(ℝ) of (*) satisfies the inequalityWe study the following problem on the strengthening estimate (**). Let a non-negative functionbe given. We have to find minimal additional restrictions forθunder which the following inequality holds:Here,yis a solution of (*) from the classLp(ℝ), andc2(p) is an absolute positive constant.

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