Semiexact 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:math> are illustrated graphically for a given potential parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow></mml:math>. We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:math> increases. This arises from the fact that the parity, which is a defined symmetry for very small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:math>, is completely violated for large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:math>. We also notice that the energy levels <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:msub><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> decrease with the increasing potential parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:math>.