Four five‐parametric and five four‐parametric independent confluent Heun potentials for the stationary Klein‐Gordon equation

Abstract

We present in total fifteen potentials for which the stationary Klein-Gordon equation is solvable in terms of the confluent Heun functions. Because of the symmetry of the confluent Heun equation with respect to the transposition of its regular singularities, only nine of the potentials are independent. Four of these independent potentials are five-parametric. One of them possesses a four-parametric ordinary hypergeometric sub-potential, another one possesses a four-parametric confluent hypergeometric sub-potential, and one potential possesses four-parametric sub-potentials of both hypergeometric types. The fourth fiveparametric potential has a three-parametric confluent hypergeometric sub-potential, which is, however, only conditionally integrable. The remaining five independent Heun potentials are four-parametric and have solutions only in terms of irreducible confluent Heun functions.

Generalized confluent hypergeometric solutions of the Heun confluent equation

“…The Heun confluent equation [1][2][3] is a second order linear differential equation widely encountered in contemporary physics research ranging from hydrodynamics, polymer and chemical physics to atomic and particle physics, theory of black holes, general relativity and cosmology, etc. (see, e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and references therein). This equation has two regular singularities conventionally located at points 0 z  and 1 z  of complex z -plane, and an irregular singularity of rank 1 at z   .…”

Exactly-Solvable Quantum Systems in Terms of Lambert-W Functions

“…It is remarkable that even though the first potential admits solutions of bound-state type, the confluent hypergeometric functions contained in those bound-state wave-functions do not degenerate to polynomials. Let us also mention that the above results on Schrödinger systems were generalized to the Klein-Gordon equation [10]. The purpose of the present work is to generalize the results presented in [8] [9] [10].…”

A conditionally exactly solvable generalization of the inverse square root potential

“…The application of the equations obeyed by the derivatives of the five Heun equations to the Schrödinger equation as well as to other analogous quantum or classical equations is rather productive. Supporting this observation is the derivation of several new exactly or conditionally exactly solvable potentials recently presented by several authors for different equations including the relativistic Dirac and Klein-Gordon equations [21][22][23][24]. Based upon the recent experience in treating the quantum two-state problem by this approach [25][26][27] we may expect more such results if a systematic study is undertaken.…”