arXiv subjects

M. E. H. Ismail

Publications and source records attributed to M. E. H. Ismail.

7 recordsLinked to original sources

arXiv2022

Solutions of the scattering problem in a complete set of Bessel functions with a discrete index

We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Kratzer potential. We do the same for a radial power-law potential with inverse-square and inverse-cube singularities. These solutions are written as infinite convergent series of Bessel functions with a discrete index. As physical application of the latter solution, we treat electron scattering off a neutral molecule with electric dipole and electric quadrupole moments.

quant-ph↗

arXiv2015

Quantum mechanics without potential function

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wave function, bound states energy spectrum and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and there-parameter systems.

quant-ph↗

arXiv2012

The Dirac-Coulomb Problem: a mathematical revisit

We obtain a symmetric tridiagonal matrix representation of the Dirac-Coulomb operator in a suitable complete square integrable basis. Orthogonal polynomials techniques along with Darboux method are used to obtain the bound states energy spectrum, the relativistic scattering amplitudes and phase shifts from the asymptotic behavior of the polynomial solutions associated with the resulting three-term recursion relation.

math-ph↗

arXiv2004

Bethe Ansatz Equations of XXZ Model and q-Sturm-Liouville Problems

In this article we have discovered a close relationship between the (algebraic) Bethe Ansatz equations of the spin $s$ XXZ model of a finite size and the $q$-Sturm-Liouville problem. We have demonstrated that solutions of the Bethe Ansatz equations give rise to the polynomial solutions of a second order $q$-difference equation in terms of Askey-Wilson operator. The more general form of Bethe Ansatz equations and the mathematical problems relevant to the physics of XXZ model are discussed. Furthermore, the similar correspondence between Bethe Ansatz equations of XXX model and the Sturm-Liouville type difference equation in terms of Wilson operator has also been found.

math-ph↗

arXiv2004

Generalizations of Chebyshev Polynomials and Polynomial Mappings

We show how polynomial mappings of degree k from a union of disjoint intervals onto [-1,1] generate a countable number of special cases of a certain generalization of the Chebyshev Polynomials. We also derive a new expression for these generalized Chebyshev Polynomials for any number of disjoint intervals from which the coefficients of x^n can be found explicitly in terms of the end points and the recurrence coefficients. We find that this representation is useful for specializing to the polynomial mapping cases for small k where we will have algebraic expressions for the recurrence coefficients in terms of the end points. We study in detail certain special cases of the polynomials for small k and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an expression for the discriminant for the case of two intervals that is valid for any configuration of the end points.

math.CA↗

arXiv2001

q-Random Matrix Ensembles

Theory of Random Matrix Ensembles have proven to be a useful tool in the study of the statistical distribution of energy or transmission levels of a wide variety of physical systems. We give an overview of certain q-generalizations of the Random Matrix Ensembles, which were first introduced in connection with the statistical description of disordered quantum conductors.

cond-mat.dis-nn↗

arXiv1995

Impact of localization on Dyson's circular ensemble

A wide variety of complex physical systems described by unitary matrices have been shown numerically to satisfy level statistics predicted by Dyson's circular ensemble. We argue that the impact of localization in such systems is to provide certain restrictions on the eigenvalues. We consider a solvable model which takes into account such restrictions qualitatively and find that within the model a gap is created in the spectrum, and there is a transition from the universal Wigner distribution towards a Poisson distribution with increasing localization.

cond-mat↗