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Bartolomeu D. B. Figueiredo

Publications and source records attributed to Bartolomeu D. B. Figueiredo.

8 recordsLinked to original sources
arXiv2021

Solutions of Heun's general equation and elliptic Darboux equation

New solutions for the elliptic Darboux equation are obtained as particular cases of solutions constructed for Heun's general equation. We consider two groups of power series expansions and two new groups of expansions in series of Gauss hypergeometric functions. The convergence of one group in power series is determined by means of ratio tests for infinite series, while the other groups are designed to solve problems which admit finite-series solutions. Actually, we envisage periodic quasi-exactly solvable potentials for which the stationary one-dimensional Schrödinger equation is reduced to the Darboux equation. In general, finite- and infinite-series solutions are obtained from power series expansions for Heun's equation. However, we show that the Schrödinger equation admits additional finite-series expansions in terms of hypergeometric functions for a family of associated Lamé potentials used in band theory of solids. For each finite-series solution we find as well four infinite-series expansions which are bounded and convergent for all values of the independent variable. Finally we find that it is possible to use transformations of variables in order to generate new solutions for the Darboux equation out of other expansions in series of hypergeometric functions for the Heun equation.

math-ph↗
arXiv2015

Confluent and Double-Confluent Heun Equations: Convergence of Solutions in Series of Coulomb Wavefunctions

The Leaver solutions in series of Coulomb wave functions for the confluent Heun equation (CHE) are given by two-sided infinite series, that is, by series where the summation index $n$ runs from minus to plus infinity [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. First we show that, in contrast to the D'Alembert test, under certain conditions the Raabe test assures that the domains of convergence of these solutions include an additional singular point. Further, by using a limit proposed by Leaver, we obtain solutions for the double-confluent Heun equation (DCHE). In addition, we get solutions for the so-called Whittaker-Ince limit of the CHE and DCHE. For these four equations, new solutions are generated by transformations of variables. In the second place, for each of the above equations we obtain one-sided series solutions ($n\geq 0$) by truncating on the left the two-sided series. Finally we discuss the time dependence of the Klein-Gordon equation in two cosmological models and the spatial dependence of the Schrödinger equation to a family of quasi-exactly solvable potentials. For a subfamily of these potentials we obtain infinite-series solutions which converge and are bounded for all values of the independent variable, in opposition to a common belief}.

math-ph↗
arXiv2015

Integral relations for solutions of confluent Heun equations

Firstly, we construct kernels of integral relations among solutions of the confluent Heun equation (CHE) and its limit, the reduced CHE (RCHE). In both cases we generate additional kernels by systematically applying substitutions of variables. Secondly, we establish integral relations between known solutions of the CHE that are power series and solutions that are series of special functions; and similarly for solutions of the RCHE. Thirdly, by using one of the integral relations as an integral transformation we obtain a new series solution of the spheroidal wave equation. From this solution we construct new solutions of the general CHE, and show that these are suitable for solving the radial part of the two-center problem in quantum mechanics. Finally, by applying a limiting process to kernels for the CHEs we obtain kernels for {two} double-confluent Heun equations. As a result, we deal with kernels of four equations of the Heun family, each equation presenting a distinct structure of singularities. In addition, we find that the known kernels for the Mathieu equation are special instances of kernels of the RCHE.

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arXiv2011

On Certain Solutions for Confluent and Double-Confluent Heun Equations

This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. Finally, we present problems which are ruled by each of these four equations and establish relations among Heun equations and quasi-exactly solvable problems.

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arXiv2011

Transformations of Heun's equation and its integral relations

We find transformations of variables which preserve the form of the equation for the kernels of integral relations among solutions of the Heun equation. These transformations lead to new kernels for the Heun equation, given by single hypergeometric functions (Lambe-Ward-type kernels) and by products of two hypergeometric functions (Erdélyi-type). Such kernels, by a limiting process, also afford new kernels for the confluent Heun equation.

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arXiv2009

A limit of the confluent Heun equation and the Schroedinger equation for an inverted potential and for an electric dipole

We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schrödinger equation with an inverted quasi-exactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole. For the first problem we find finite- and infinite-series solutions which are convergent and bounded for any value of the independent variable. For the angular equation, we also find expansions in series of Jacobi polynomials.

math-ph↗
arXiv2004

Integral relations and new solutions to the double-confluent Heun equation

Integral relations and transformation rules are used to obtain, out of an asymptotic solution, a new group of four pairs of solutions to the double-confluent Heun equation. Each pair presents the same series coefficients but has solutions convergent in different regions of the complex plane. Integral relations are also established between solutions given by series of Coulomb wave functions. The Whittaker-Hill equation and another equation are studied as particular cases of both the double-confluent and the single-confluent Heun equations. Finally, applications for the Schrödinger equation with certain potentials are discussed, mainly for quasi-exactly solvable potentials which lead to the above special equations.

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