Integral relations for solutions of the confluent Heun equation

Abstract: Firstly, we construct kernels for integral relations among solutions of the confluent Heun equation (CHE). Additional kernels are systematically generated by 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. Thirdly, by using one of the integral relations as an integral transformation we obtain a new series solution of the ordinary spheroidal wave equation (a particular CHE). ¿From this solution we construct new series solutions of the general CHE, and show that these are suitable for solving the radial part of the two-center problem in quantum mechanics. 1. Introductory remarks Recently we have found that the transformations of variables which preserve the form of the general Heun equation correspond to transformations which preserve the form of the equation for kernels of integral relations among solutions of the Heun equation [1]. In fact, by using the known transformations of the Heun equation [2, 3] we have found prescriptions for transforming kernels and, in this manner, we have generated several new kernels for the equation. The above correspondence can be extended to the confluent equations of the Heun family, that is, to the (single) confluent, double-confluent, biconfluent and triconfluent Heun equations [4, 5], as well as to the reduced forms of such equations [6, 7]. In the present study we investigate only the confluent Heun equation (CHE). Specifically: • we deal with the construction and transformations of integral kernels for the CHE; • from some of these kernels, we establish integral relations between known solutions of the CHE; • from another kernel, we obtain new solutions in series of confluent hypergeometric functions for the CHE; we show that these solutions are suitable to solve the radial part of the two-centre problem in quantum mechanics [8]. We write the CHE as [9] z(z − z0) dU dz2 + (B1 +B2z) dU dz + [ B3 − 2ωη(z − z0) + ωz(z − z0) ] U = 0, [CHE] (1) where z0, Bi, η and ω are constants. This CHE is called generalized spheroidal wave equation by Leaver [9], but sometimes the last expression refers to a particular case of the CHE [5, 10]. Excepting the Mathieu equation, the CHE is the most studied of the confluent Heun equations and includes the (ordinary) spheroidal equation [5] as a particular case. Its recent occurrence in several classes of quantum two-state systems [11], certainly will require new solutions for Eq. (1). On the other side, the reduced confluent Heun equation (RCHE) is written as z(z − z0) dU dz2 + (B1 +B2z) dU dz + [B3 + q(z − z0)]U = 0, [RCHE] (2) where z0, Bi and q (q 6= 0) are constants. The RCHE describe the angular part of the Schrödinger equation for an electron in the field of a point electric dipole [12, 13]. It appears as well in the study of two-level systems [14], polymer dynamics [15] and theory of gravitation [16]. The form (2) for the RCHE results from the CHE (1) by means of the limits ω → 0, η →∞ such that 2ηω = −q, [Whittaker-Ince limit]. (3) In both equations, z = 0 and z = z0 are regular singular points with exponents (0, 1 + B1/z0) and (0, 1 − B2 − B1/z0), respectively, that is, from ascending power series solutions we find lim z→0 U(z) ∼ 1, lim z→0 U(z) ∼ z B1 z0 ; lim z→z0 U(z) ∼ 1, lim z→z0 U(z) ∼ (z − z0)2 B1 z0 : [CHE, RCHE]. (4) Electronic address: [email protected] Electronic address: [email protected] CBPF-NF001/15