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.

A fourth order limit-3 expression with nonempty essential spectrum

1987 ◽  
Vol 106 (3-4) ◽  
pp. 233-235 ◽  
Author(s):  
Bernd Schultze
Keyword(s):  
Essential Spectrum ◽  
Fourth Order ◽  
Limit Point ◽  
Order Case ◽  
Limit Point Case

SynopsisIt is shown that even in the fourth order case there exist real symmetric ordinary differential expressions with nonempty essential spectrum which are not in the limit-point case.

A limit-point criterion for 2n-th order symmetric differential expressions

1981 ◽  
Vol 90 (1-2) ◽  
pp. 1-12 ◽  
Author(s):  
Bernd Schultze
Keyword(s):  
Limit Point ◽  
The Other ◽  
Perturbation Theorem ◽  
Deficiency Indices ◽  
Rate Of Growth ◽  
Even Order ◽  
The Absolute

SynopsisA perturbation theorem for symmetric differential expressions is given and applied to even-order expressions. We assume the coefficients p0(t), pn(t) to be eventually positive and to have real powers of t as dominating terms. Then we are able to admit for the absolute values of the other coefficients a rate of growth depending on the growth of both of the coefficients p0, pn in order to obtain minimal deficiency indices.

On the stability of the limit-point property of “Kauffman expressions” under relatively bounded perturbations

1986 ◽  
Vol 103 (1-2) ◽  
pp. 73-89 ◽  
Author(s):  
Bernhard Mergler ◽  
Bernd Schultze
Keyword(s):  
Perturbation Theory ◽  
Limit Point ◽  
Point Property ◽  
Limit Point Case ◽  
Whole Class ◽  
The Stability ◽  
Bounded Perturbations ◽  
Positive Coefficients

SynopsisIn this paper we show that for the whole class of differential expressions in the limit-point case considered by Kauffman in [2], the perturbation theory yields a limit-point criterion for a much wider class of ordinary differential expressions. More general coefficients are admitted which may be eventually negative provided they are “dominated” by some other positive coefficients. This generalises results in [4], [5] and [6].

scholarly journals Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yan Liu ◽  
Meiru Xu
Keyword(s):  
Perturbation Theory ◽  
Hamiltonian Systems ◽  
Limit Point ◽  
Limit Circle ◽  
Linear Relations ◽  
Deficiency Indices ◽  
Bounded Perturbations

AbstractThis paper is concerned with stability of deficiency indices for discrete Hamiltonian systems under perturbations. By applying the perturbation theory of Hermitian linear relations we establish the invariance of deficiency indices for discrete Hamiltonian systems under bounded perturbations. As a consequence, we obtain the invariance of limit types for the systems under bounded perturbations. In particular, we build several criteria of the invariance of the limit circle and limit point cases for the systems. Some of these results improve and extend some previous results.

scholarly journals Influence of Even-Order Dispersion on Super-Sech Soliton Transmission Quality under Coherent Crosstalk

2008 ◽  
Vol 2008 ◽  
pp. 1-5 ◽  
Author(s):  
Aleksandra Panajotovic ◽  
Daniela Milovic ◽  
Anjan Biswas ◽  
Essaid Zerrad
Keyword(s):  
Fourth Order ◽  
Optical Network ◽  
Single Mode ◽  
Single Mode Fiber ◽  
Higher Order ◽  
Trailing Edges ◽  
Even Order ◽  
Transmission Quality ◽  
The Impact ◽  
Order Dispersion

The transmission speed of optical network strongly depends on the impact of higher order dispersion. In presence of coherent crosstalk, which cannot be otherwise controlled by optical filtering, the impact of higher order dispersions becomes more pronounced. In this paper, the general expressions, that describe pulse deformation due to second- and fourth-order dispersions in a single-mode fiber, are given. The responses for such even-order dispersions, in presence of coherent crosstalk, are characterized by waveforms with long trailing edges. The transmission quality of optical pulses, due to both individual and combined influence of second- and fourth-order dispersions, is studied in this paper. Finally, the pulse shape and eye diagrams are obtained.

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.

The Oscillation of Fourth Order Linear Differential Operators

1975 ◽  
Vol 27 (1) ◽  
pp. 138-145 ◽  
Author(s):  
Roger T. Lewis
Keyword(s):  
Fourth Order ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Oscillatory Behavior ◽  
The Self ◽  
Order Operator ◽  
Order Linear ◽  
Linear Differential Operators ◽  
Even Order ◽  
Linear Differential

Define the self-adjoint operatorwhere r(x) > 0 on (0, ∞) and q and p are real-valued. The coefficient q is assumed to be differentiate on (0, ∞) and r is assumed to be twice differentia t e on (0, ∞).The oscillatory behavior of L4 as well as the general even order operator has been considered by Leigh ton and Nehari [5], Glazman [2], Reid [7], Hinton [3], Barrett [1], Hunt and Namb∞diri [4], Schneider [8], and Lewis [6].

scholarly journals Analyzing “Homotopy Perturbation Method for Solving Fourth-Order Boundary Value Problem”

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Afgan Aslanov
Keyword(s):  
Boundary Value Problem ◽  
Perturbation Method ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Efficient Method ◽  
Homotopy Perturbation Method ◽  
Boundary Value ◽  
Homotopy Perturbation

We analyze a previous paper by S. T. Mohyud-Din and M. A. Noor (2007) and show the mistakes in it. Then, we demonstrate a more efficient method for solving fourth-order boundary value problems.

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