scholarly journals The limit-point and limit-circle classification of the Sturm-Liouville operator (py′)′ + qy

1974 ◽  
Vol 10 (2) ◽  
pp. 315-316
Author(s):  
Ian Walker Knowles
Keyword(s):  
Limit Point ◽  
Liouville Operator ◽  
Limit Circle ◽  
Sturm Liouville

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.

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.

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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