scholarly journals SPECTRAL PROPERTIES OF OPERATORS USING TRIDIAGONALIZATION

2012 ◽  
Vol 10 (03) ◽  
pp. 327-343 ◽  
Author(s):  
MOURAD E. H. ISMAIL ◽  
ERIK KOELINK
Keyword(s):  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Difference Operator ◽  
General Scheme ◽  
Second Order ◽  
Order Differential Operator ◽  
Difference Operators ◽  
Jacobi Function ◽  
Wilson Polynomials ◽  
Tridiagonal Form

A general scheme for tridiagonalizing differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure of generally different orthogonal polynomials. Three examples are worked out: (1) related to Jacobi and Wilson polynomials for a second order differential operator, (2) related to little q-Jacobi polynomials and Askey–Wilson polynomials for a bounded second order q-difference operator, (3) related to little q-Jacobi polynomials for an unbounded second order q-difference operator. In case (1) a link with the Jacobi function transform is established, for which we give a q-analogue using example (2).

Limit Transitions for BC Type Multivariable Orthogonal Polynomials

1997 ◽  
Vol 49 (2) ◽  
pp. 374-405 ◽  
Author(s):  
Jasper V. Stokman ◽  
Tom H. Koornwinder
Keyword(s):  
Orthogonal Polynomials ◽  
Root System ◽  
Jacobi Polynomials ◽  
Parameter Family ◽  
Difference Operators ◽  
Wilson Polynomials ◽  
The One

AbstractLimit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are q-analogues of the three parameter family of BC type Jacobi polynomials in n variables. The limit transitions will be derived by taking limits of q-difference operators which have these polynomials as eigenfunctions.

Non-self-adjoint difference operators and their spectrum

2005 ◽  
Vol 461 (2057) ◽  
pp. 1505-1531 ◽  
Author(s):  
Robert Howard Wilson
Keyword(s):  
Difference Operator ◽  
Discrete Analogue ◽  
Second Order ◽  
Difference Operators ◽  
Essential Difference ◽  
Order Difference ◽  
Second Order Differential Equations ◽  
Forward Difference ◽  
Difference Expression ◽  
Complex Coefficients

Initially, this paper is a discrete analogue of the work of Brown et al. (1999 Proc. R. Soc. A 455 , 1235–1257) on second-order differential equations with complex coefficients. That is, we investigate the general non-self-adjoint second-order difference expression where the coefficients p n and q n are complex and Δ is the forward difference operator, i.e. Δ x n = x n +1 − x n . Properties of the so-called Hellinger–Nevanlinna m -function for the recurrence relation Mx n = λ w n x n , where the w n are real and positive, are examined, and relationships between the properties of the m -function and the spectrum of the associated operator are explored. However, an essential difference between the continuous and the discrete case arises in the way in which we define the operator natural to the problem. Nevertheless, analogous results regarding the spectrum of this operator are obtained.

Orlicz lacunary sequence spaces of 𝑙-fractional difference operators

2020 ◽  
Vol 26 (2) ◽  
pp. 173-183
Author(s):  
Kuldip Raj ◽  
Kavita Saini ◽  
Anu Choudhary
Keyword(s):  
Difference Operator ◽  
Lacunary Sequence ◽  
Sequence Spaces ◽  
Difference Operators ◽  
Normed Spaces ◽  
Fractional Difference ◽  
New Approach ◽  
Inclusion Relations ◽  
Difference Sequence Spaces ◽  
Bounded Sequences

AbstractRecently, S. K. Mahato and P. D. Srivastava [A class of sequence spaces defined by 𝑙-fractional difference operator, preprint 2018, http://arxiv.org/abs/1806.10383] studied 𝑙-fractional difference sequence spaces. In this article, we intend to make a new approach to introduce and study some lambda 𝑙-fractional convergent, lambda 𝑙-fractional null and lambda 𝑙-fractional bounded sequences over 𝑛-normed spaces. Various algebraic and topological properties of these newly formed sequence spaces have been explored, and some inclusion relations concerning these spaces are also established. Finally, some characterizations of the newly formed sequence spaces are given.

Stability of Solutions of Differential-operator and Operator-difference Equations in the Sense of Perturbation of Operators

2006 ◽  
Vol 6 (3) ◽  
pp. 269-290 ◽  
Author(s):  
B. S. Jovanović ◽  
S. V. Lemeshevsky ◽  
P. P. Matus ◽  
P. N. Vabishchevich
Keyword(s):  
Differential Operator ◽  
Difference Equations ◽  
Hyperbolic Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Order Differential Operator ◽  
Stability Of Solutions ◽  
Operator Equations ◽  
The Difference ◽  
Coefficient Stability

Abstract Estimates of stability in the sense perturbation of the operator for solving first- and second-order differential-operator equations have been obtained. For two- and three-level operator-difference schemes with weights similar estimates hold. Using the results obtained, we construct estimates of the coefficient stability for onedimensional parabolic and hyperbolic equations as well as for the difference schemes approximating the corresponding differential problems.

scholarly journals Exact solvability of two new 3D and 1D non-relativistic potentials within the TRA framework

2018 ◽  
Vol 33 (32) ◽  
pp. 1850187 ◽  
Author(s):  
I. A. Assi ◽  
H. Bahlouli ◽  
A. Hamdan
Keyword(s):  
Wave Function ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Basis Functions ◽  
Exact Solvability ◽  
Analytical Properties ◽  
Expansion Coefficients ◽  
Potential Parameters ◽  
Square Integrable Basis

This work aims at introducing two new solvable 1D and 3D confined potentials and present their solutions using the Tridiagonal Representation Approach (TRA). The wave function is written as a series in terms of square integrable basis functions which are expressed in terms of Jacobi polynomials. The expansion coefficients are then written in terms of new orthogonal polynomials that were introduced recently by Alhaidari, the analytical properties of which are yet to be derived. Moreover, we have computed the numerical eigenenergies for both potentials by considering specific choices of the potential parameters.

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