• Corpus ID: 115617900

New solutions to the confluent Heun equation and quasiexact solvability

@article{ElJaick2013NewST,
  title={New solutions to the confluent Heun equation and quasiexact solvability},
  author={L{\'e}a Jaccoud El-Jaick and Bartolomeu D. B. Figueiredo},
  journal={arXiv: Mathematical Physics},
  year={2013}
}
We construct new solutions in series of confluent hypergeometric functions for the confluent Heun equation (CHE). Some of these solutions are applied to the one-dimensional stationary Schr\"{o}dinger equation with hyperbolic and trigonometric quasiexactly solvable potentials. 
Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions
We examine the series expansions of the solutions of the confluent Heun equation in terms of three different sets of the Kummer confluent hypergeometric functions. The coefficients of the expansions
Appell Hypergeometric Expansions of the Solutions of the General Heun Equation
Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun
SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS
  • A. Ishkhanyan
  • Mathematics
    Journal of Applied Analysis & Computation
  • 2019
We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having
The Floquet Theory of the Two-Level System Revisited
  • H. Schmidt
  • Mathematics, Physics
    Zeitschrift für Naturforschung A
  • 2018
Abstract In this article, we reconsider the periodically driven two-level system especially the Rabi problem with linear polarisation. The Floquet theory of this problem can be reduced to its
Fifteen classes of solutions of the quantum two-state problem in terms of the confluent Heun function
We derive 15 classes of time-dependent two-state models solvable in terms of the confluent Heun functions. These classes extend over all the known families of three- and two-parametric models

References

SHOWING 1-10 OF 21 REFERENCES
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
Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions
The Leaver solutions in series of Coulomb wavefunctions for the confluent Heun equation are given by two-sided infinite series, that is, by series where the summation index n runs from minus to plus
Integral relations for heun-class special functions
New integral relations are obtained for eigenfunctions produced by Heun-class equations. These relations demonstrate the duality property of the eigenfunctions with different behaviors at
Ince’s limits for confluent and double-confluent Heun equations
We find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation (this is also known as confluent Heun equation). One solution in
An exactly soluble Schrödinger equation with a bistable potential
For a bistable potential which is the sum of two hyperbolic cosine functions, the Schrodinger equation for the low‐lying states of a homonuclear diatomic molecule can be solved analytically. In this
Exact and quasiexact solvability of second order superintegrable quantum systems. II. Relation to separation of variables
We make explicit the intimate relationship between quasiexact solvability, as expounded, for example, by Ushveridze [Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1993)], and the
XXI—The Whittaker-Hill Equation and the Wave Equation in Paraboloidal Co-ordinates
  • F. Arscott
  • Mathematics
    Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences
  • 1967
Synopsis The problem considered is that of obtaining solutions of the Helmholz equation ∇2V + k2V = 0, suitable for use in connection with paraboloidal co-ordinates. In these co-ordinates the
Generalized spheroidal wave equation and limiting cases
We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries
We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum
Analytic Solutions of the Teukolsky Equation and Their Low Frequency Expansions
Analytic solutions of the Teukolsky equation in Kerr geometries are presented in the form of series of hypergeometric functions and Coulomb wave functions. Relations between these solutions are
...
...