Special Functions Of Mathematical Physics Research Articles

Soliton-like solutions for the nonlinear schrödinger equation with variable quadratic hamiltonians

We construct one-soliton solutions for the nonlinear Schr¨odinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of the complete (super) integrability of generalized harmonic oscillators. The soliton-wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schr¨odinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painlev´e transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons and Jacobi elliptic and second Painlev´e transcendental solutions for several variable Hamiltonians that are important for research in nonlinear optics, plasma physics, and Bose–Einstein condensation. The Feshbach-resonance matter-wave-soliton management is briefly discussed from this new perspective.

Table Errata to “Formulas and theorems for the special functions of mathematical physics” by W. Magnus, F. Oberhettinger & R. P. Soni (1966)

Table Errata to “Formulas and theorems for the special functions of mathematical physics” by W. Magnus, F. Oberhettinger & R. P. Soni (1966)

On eigenvalue problems in quantum mechanics

To solve quantum mechanical eigenvalue problems using the algorithmic methods recently derived by Nikiforov and Uvarov (1988 Special Functions of Mathematical Physics (Basel: Birkhäuser)) and Ciftci et al (2003 J. Phys. A: Math. Gen. 36 11807), one needs to first convert the associated wave equation into hypergeometric or closely related forms. We point out that once such forms are obtained, the eigenvalue problem can be satisfactorily solved by only imposing the condition that the regular infinite series solutions of the equations should become polynomials, and one need not take recourse to the use of the algorithmic methods. We first demonstrate the directness and simplicity of our approach by dealing with a few case studies and then present new results for the Woods–Saxon potential.

Laplace-type equations as conformal superintegrable systems

We lay out the foundations of the theory of second order conformal superintegrable systems. Such systems are essentially Laplace equations on a manifold with an added potential: (Δn+V(x))Ψ=0. Distinct families of second order superintegrable Schrödinger (or Helmholtz) systems (Δn′+V′(x))Ψ=EΨ can be incorporated into a single Laplace equation. There is a deep connection between most of the special functions of mathematical physics, these Laplace conformally superintegrable systems and their conformal symmetry algebras. Using the theory of the Laplace systems, we show that the problem of classifying all 3D Helmholtz superintegrable systems with nondegenerate potentials, i.e., potentials with a maximal number of independent parameters, can be reduced to the problem of classifying the orbits of the nonlinear action of the conformal group on a 10-dimensional manifold.

Special Function Inverse Series Pairs

Working with the various special functions of mathematical physics and applied mathematics we often encounter inverse relations of the type and , where 0, 1, 2,. Here , denote special polynomial functions, and , denote coefficients found by use of the orthogonal properties of and , or by skillful series manipulations. Typically and , the n-th Legendre polynomial. We give a collection of inverse series pairs of the type if and only if , each pair being based on some reasonably well-known special function. We also state and prove an interesting generalization of a theorem of Rainville in this form.

Gravitational wave signal templates, pattern recognition, and reciprocal Eulerian gamma functions

The direct detection of Gravitational Waves (GWs) is one of the most challenging problems in experimental gravitation today. It necessitates the use of highly advanced large laser interferometers such as LIGO, VIRGO, LISA, TAMA 300, GEO 600 and AIGO. The analysis of the data from such instruments requires and combines the expertise from a multitude of scientific disciplines. The verification of a detected signal demands an effective way to distinguish the source signal from the background noise. Such a study is required for an all-sky search to determine the ϕ and θ angles on the sky of gravitational wave sources and their frequencies. In this paper, we present analytical solutions and associated numerical approximations for the inner products employed in matched filtering a GW signal using templates. An exact closed-form expression for the inner products is rigourously derived using the special functions of mathematical physics. The inner products involve reciprocal Eulerian gamma functions, which occur in the study of many diverse phenomena. The spectral noise density of the VIRGO GW detector is shown to be amenable to our analysis. Spectral noise densities like those for LIGO and GEO 600, although different and in a slightly more restricted frequency band, are likewise amenable. We study numerical computation of the inner products, estimate the computational time of the solution on serial and parallel computers, and show the efficiency of the resulting algorithms. The fitting factor that indicates the goodness of fit between a signal and a template is given in closed-form and computed numerically. The numerical plots display an approximate symmetry in the template ϕ and θ domain.

Fisher information of special functions and second-order differential equations

We investigate a basic question of analytic information theory, namely, the evaluation of the Fisher information and the relative Fisher information with respect to a non-negative function, for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second-order differential equations. We obtain explicit expressions for these information-theoretic properties via the expectation values of the coefficients of the differential equation. We illustrate our approach for various nonrelativistic D-dimensional wavefunctions and some special functions of physicomathematical interest. Emphasis is made in the Nikiforov–Uvarov hypergeometric-type functions, which include and generalize the Hermite functions and the Gauss and Kummer hypergeometric functions, among others.

On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions

Perturbative quantum field theory for the Ising model at the three-loop level yields a tetrahedral Feynman diagram C(a,b) with masses a and b and four other lines with unit mass. The completely symmetric tetrahedron CTet≡C(1,1) has been of interest from many points of view, with several representations and conjectures having been given in the literature. We prove a conjectured exponentially fast convergent sum for C(1,1), as well as a previously empirical relation for C(1,1) as a remarkable difference of Clausen function values. Our presentation includes propositions extending the theory of the dilogarithm Li2 and Clausen Cl2 functions, as well as their relation to other special functions of mathematical physics. The results strengthen connections between Feynman diagram integrals, volumes in hyperbolic space, number theory, and special functions and numbers, specifically including dilogarithms, Clausen function values, and harmonic numbers.

Multiparameter spectral theory and separation of variables

Multiparameter spectral theory generalizes the classical spectral theory of linear operators to n linear operators linked by n spectral parameters. The spectral parameters arise as separation constants when, for example, separation of variables techniques are used to solve boundary-value problems for partial differential equations. In this paper, we discuss some of the fundamental aspects of the theory such as solvability, commutativity and abstract representation theory. Many of these concepts are of importance to modern developments in separation of variables techniques. The ideas are illustrated in application to differential equations and special functions of mathematical physics.

P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation

The solution of several instances of the Schrodinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585–600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Grunbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461–84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.

Some Representations of Unified Voigt Functions

The authors derive a set of unified representations of the Voigt functions in terms of familiar special functions of Mathematical Physics. Some deductions from these representations are also considered.

Fractional operators and special functions. I. Bessel functions

Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements D of the associated Lie algebras as linear differential operators gives relations among the functions in a class, for example, their differential recurrence relations. In this paper, we define fractional generalizations Dμ of these operators in the context of Lie theory, determine their formal properties, and illustrate their use in obtaining interesting relations among the functions. We restrict our attention here to the Euclidean group E(2) and the Bessel functions. We show that the two-variable fractional operator relations lead directly to integral representations for the Bessel functions, reproduce known fractional integrals for those functions when reduced to one variable, and contribute to a coherent understanding of the connection of many properties of the functions to the underlying group structure. We extend the analysis to the associated Legendre functions in a following paper.

Fractional operators and special functions. II. Legendre functions

Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements D of the associated Lie algebras as linear differential operators gives relations among the functions in a class, for example, their differential recurrence relations. In this paper, we apply the fractional generalizations Dμ of these operators developed in an earlier paper in the context of Lie theory to the group SO(2,1) and its conformal extension. The fractional relations give a variety of interesting relations for the associated Legendre functions. We show that the two-variable fractional operator relations lead directly to integral relations among the Legendre functions and to one- and two-variable integral representations for those functions. Some of the relations reduce to known fractional integrals for the Legendre functions when reduced to one variable. The results enlarge the understanding of many properties of the associated Legendre functions on the basis of the underlying group structure.

Exponential Type Complex and Non-Hermitian Potentials in PT-Symmetric Quantum Mechanics

Using the NU method [A F Nikiforov and V B Uvarov 1988 “Special Functions of Mathematical Physics” (Birkhauser, Basel)], we investigated the real eigenvalues of the complex and/or PT-symmetric, non-Hermitian and the exponential type systems, such as Pöschl–Teller and Morse potentials.

Canonical transformations and non-unitary evolution

We test the idea that transformations which, at the classical level, can be interpreted as evolutions are represented within quantum mechanics by unitary operators. To this end, we consider non-trivial canonical transformations which leave invariant the form of the Hamilton function of a system. We demonstrate that infinite families of such transformations exist for a variety of familiar conservative systems of one degree of freedom. We show how the precise form of integral equations for the stationary state wavefunctions implied by the existence of these canonical transformations can be pinned down by exploiting the algebra of the transformations and a symmetry of their generating functions. We recover several integral equations found in the literature on standard special functions of mathematical physics. We find that when one of the classical canonical transformations we consider is non-linear, its quantum implementation is non-unitary. We end with some comments on the implications of our findings for semiclassical studies and a brief discussion relevant to string theory of the generalization to scalar field theories in 1 + 1 dimensions.

First order stochasticity in boundary geometry: a computer algebra BE development

Evaluation of boundary element responses due to a small change in the system data always involves elaborate differentiation of complicated expressions associated with inverses of matrices containing elementary and special functions of mathematical physics. Such calculations are essential in reliability analysis of structural and mechanical systems, particularly in aerospace engineering, and bioengineering modeling, for which high quality solutions are necessary. In a computer algebra environment, modification of multiplication and power expansion rules is sufficient for automatic implementation of first order perturbation in an existing boundary element package written for deterministic variables. The closed form algebraic manipulations are independent of the problem size.

Exact solutions of the Schrödinger equation for the deformed hyperbolic potential well and the deformed four-parameter exponential type potential

Exact solutions of the Schrödinger equation for two `deformed hyperbolic potentials' that were introduced by Arai [J. Math. Anal. Appl. 158 (1991) 63] have been obtained by using an analytical solution method which has been developed by Nikiforov and Uvarov [Special Functions of Mathematical Physics, Birkhauser, Basel, 1988] for the solutions of the differential equations of hypergeometric type in which the solutions are special functions.

Structural theory of special functions

A block diagram is suggested for classifying differential equations whose solutions are special functions of mathematical physics. Three classes of these equations are identified: the hypergeometric, Heun, and Painleve classes. The constituent types of equations are listed for each class. The confluence processes that transform one type into another are described. The interrelations between the equations belonging to different classes are indicated. For example, the Painleve-class equations are equations of classical motion for Hamiltonians corresponding to Heun-class equations, and linearizing the Painleve-class equations leads to hypergeometric-class equations. The “confluence principle” is stated, and an example of its application is given.

Database of the special functions of mathematical physics

We describe a knowledge-based computer project on the special functions of mathematical physics, which uses a unified structural approach to their classification. A prototype program package SFTools is presented. Special functions addressed by the system include hypergeometric-type functions, Heun functions, Painlevé transcendents and orthogonal polynomials. For these functions, basic differential equations are given with emphasis on singularities, integral relations, series expansions, Riemann P-symbols (Riemann scheme) and other important characteristics. In order to structure the information, we use the processes of confluence and reduction of equations widely.

All the special functions are fractional differintegrals of elementary functions

In this survey we discuss a unified approach to the generalized hypergeometric functions based on a generalized fractional calculus developed in the monography by Kiryakova. This generalization of the classical theory of the operators of integration and differentiation of fractional order deals with integral (differintegral) operators involving Meijer's G- and Fox's H-functions as kernel functions. Their theory is fully developed and illustrated by various special cases and applications in different areas of the applicable analysis. Usually, the special functions of mathematical physics are defined by means of power series representations. However, some alternative representations can be used as their definitions. Let us mention the well known Poisson integrals for the Bessel functions and the analytical continuation of the Gauss hypergeometric function via the Euler integral formula. The Rodrigues differential formulae, involving repeated or fractional differentiation are also used as definitions of the classical orthogonal polynomials and their generalizations. As to the other special functions (most of them being - and -functions), such representations are less popular and even unknown in the general case. There exist various integral and differential formulae, but, unfortunately, they are quite peculiar for each corresponding special function and scattered in the literature without any common idea to relate them. Here, all the generalized hypergeometric functions are proved to be generalized fractional integrals or derivatives of three basic elementary functions. On this base, they are classified in three specific classes and several new integral and differential representations are found.