An algorithm to obtain global solutions of the double confluent Heun equation

Abstract

Abstract. A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schrödinger equation with a quasi-exactly-solvable potential

Global solutions of the biconfluent Heun equation

“…A procedure for obtaining the connection factors in a second-order homogeneous linear differential equation was given, forty years ago, by Naundorf [14]. An improvement of the procedure was presented in a former paper [6] and applied, in a subsequent one [1], to solutions of the double confluent Heun equation. For convenience of the reader, we recall here the foundations of the procedure, adapted to the case at hand.…”

“…Although the study of the Heun equation and its confluent forms started as early as 1888 [8], their interest, from the point of view of applications, was generally ignored until the celebration of a "Centennial Workshop" which motivated the publication of J. Sesma javier@unizar.es E. M. Ferreira erasmo@if.ufrj.br 1 Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, Brasil 2 Departamento de Física Teórica, Facultad de Ciencias, 50009, Zaragoza, Spain the well known book edited by Ronveaux [15]. An alternative study of the Heun class of equations, based on singularities, can be found in the book by Slavyanov and Lay [16].…”

“…In this paper we present a procedure which allows one to write global solutions of the BHE, i. e., solutions valid in the vicinity of both singular points of the differential equation, independently of the values of its parameters. The procedure, inspired by the treatment of the connection problem by Naundorf [14], was first proposed for solving the Schrödinger equation with a polynomial potential [6], and, conveniently modified, was used to obtain global solutions of the double confluent Heun equation [1]. We here recall the procedure, with the notation usually adopted when dealing with the BHE.…”

NumGfun

“…Finally, even the impressive range of built-in functions of computer algebra systems is not always sufficient for applications. Works on the implementation of classes of "less common" special functions that overlap those considered in NumGfun include [1,14].…”

“…Relevant algorithms exist in the literature. In particular, D-finite functions may be computed with an absolute error bounded by 2 −n in n log O (1) n bit operations-that is, in softly linear time in the size of the result written in fixed-point notation-at any point of their Riemann surfaces [12], the necessary error bounds also being computed from the differential equation and initial values [32]. However, these algorithms have remained theoretical [13, §9.2.1].…”