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 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.

The least limit point of the spectrum associated with sinǵular differential operators

1970 ◽  
Vol 67 (2) ◽  
pp. 277-281 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Operator ◽  
Linear Operator ◽  
Essential Spectrum ◽  
Limit Point ◽  
Compact Support ◽  
Complex Space ◽  
Differential Operators ◽  
Adjoint Extension ◽  
Image Position

Let τ be the formally self-adjoint differential operator denned bywhere the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.

Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators

1981 ◽  
Vol 33 (1) ◽  
pp. 229-246 ◽  
Author(s):  
Calvin D. Ahlbrandt ◽  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Differential Operators ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Dimensional Vector ◽  
Selfadjoint Extension ◽  
Image Position ◽  
Vector Valued Function ◽  
Necessary And Sufficient ◽  
Vector Valued

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l bywhere y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfyingwith inner productwhere . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

On the invariance of the essential spectrum of an arbitrary operator. III

1968 ◽  
Vol 64 (4) ◽  
pp. 975-984 ◽  
Author(s):  
Martin Schechter
Keyword(s):  
Essential Spectrum ◽  
Real Axis ◽  
Limit Point ◽  
Sufficient Conditions ◽  
Energy Operator ◽  
Hydrogen Energy ◽  
Negative Real Axis ◽  
Negative Eigenvalues ◽  
Arbitrary Operator ◽  
Image Position

The spectrum of the hydrogen energy operator(Δ is the Laplacian and r is the distance from the origin) consists of the non-negative real axis and a sequence of negative eigenvalues of finite multiplicities converging to O. In the present study we are interested in finding sufficient conditions on a potential q(x) such that the spectrum of the operatorin En has a ‘hydrogen-like’ spectrum, i.e. a spectrum consisting of(a) the non-negative real axis,(b) at most a denumerable set of negative eigenvalues of finite multiplicities having zero as its only possible limit point.

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.

Limit Point Criteria for Differential Equations, II

1974 ◽  
Vol 26 (02) ◽  
pp. 340-351 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Differential Equations ◽  
Weight Function ◽  
Limit Point ◽  
Differential Operators ◽  
Linearly Independent ◽  
Integrable Functions ◽  
Image Position ◽  
Continuous Weight ◽  
Finite Singularity

We consider here singular differential operators, and for convenience the finite singularity is taken to be zero. One operator discussed is the operator L defined by where q 0 &gt; 0 and the coefficients q t are real, locally Lebesgue integrable functions defined on an interval (a, b). For a given positive, continuous weight function h, conditions are given on the functions qi for which the number of linearly independent solutions y of L(y) = λhy (Re λ = 0) satisfying.

scholarly journals Limit Circle/Limit Point Criteria for Second-Order Superlinear Differential Equations with a Damping Term

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Lihong Xing ◽  
Wei Song ◽  
Zhengqiang Zhang ◽  
Qiyi Xu
Keyword(s):  
Differential Equations ◽  
Limit Point ◽  
Second Order ◽  
Damping Term ◽  
Limit Circle ◽  
New Criteria ◽  
Point Type ◽  
Circle Limit

The purpose of the present paper is to establish some new criteria for the classifications of superlinear differential equations as being of the nonlinear limit circle type or of the nonlinear limit point type. The criteria presented here generalize some known results in literature.

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