Semi-exact solutions of the Razavy potential

Abstract

In this work we study the quantum system with the symmetric Razavy potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun functions. The eigenvalues have to be calculated numerically. The properties of the wave functions depending on $m$ are illustrated graphically for a given potential parameter $\xi$. We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter $m$ increases. This arises from the fact that the parity, which is a defined symmetry for very small $m$, is completely violated for large $m$. We also notice that the energy levels ${\epsilon }_i$ decrease with the increasing potential parameter $m$.

10 pages, 6 figures and 1 table

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• Quasi-exactly solvable hyperbolic potential and its anti-isospectral counterpart
• Schrödinger equation with Pauli-Fierz Hamiltonian and double well potential as model of vibrationally enhanced tunneling for proton transfer in hydrogen bond