2016/01/31 by A. M. Ishkhanyan, A.M. Ishkhanyan · 56 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Basic hypergeometric series #Gauss #Generalization #Generalized hypergeometric function #Hypergeometric distribution #Hypergeometric function #Integrable system #Quantum Mechanics and Non-Hermitian Physics #Quantum and Classical Electrodynamics #Transformation (genetics) #quant-ph
paper · pdf · doi:10.1016/j.aop.2017.11.033
published in Annals of Physics 388, 456-471 (Elsevier BV)
arxiv created 2017/08/06 · openalex publication_date 2017/12/06 · arxiv updated 2017/12/22 · openalex created_date 2017/12/22 · openalex updated_date 2026/08/05
We show that there exist 35 choices for the coordinate transformation each leading to a potential for which the stationary Schrödinger equation is exactly solvable in terms of the general Heun functions. Because of the symmetry of the Heun equation with respect to the transposition of its singularities only eleven of these potentials are independent. Four of these independent potentials are always explicitly written in terms of elementary functions, one potential is given through the Jacobi elliptic sn-function, and the others are in general defined parametrically. Nine of the independent potentials possess exactly or conditionally integrable hypergeometric sub-potentials for which each of the fundamental solutions of the Schrödinger equation is written through a single hypergeometric function. Many of the potentials possess sub-potentials for which the general solution is written through fundamental solutions each of which is a linear combination of two or more Gauss hypergeometric functions. We present an example of such a potential which is a conditionally integrable generalization of the third exactly solvable Gauss hypergeometric potential.