Exact solutions of the Schrödinger equation for another class of hyperbolic potential wells

Abstract

In this work a new scheme is proposed to study the exact solutions of another class of hyperbolic potentials U p . We first obtain two linearly dependent eigenfunctions corresponding to the same even-parity state or odd-parity state by taking different variable substitutions and function transformations and then find that their solutions can be expressed analytically as the confluent Heun functions (CHFs). The Wronskian determinant which can be constructed by two linearly dependent eigenfunctions is used to get the corresponding energy spectrum equations with respect to even- and odd-parity states. According to energy spectrum equation, we can first study the intersection distribution between F(ε) which is defined by the Wronskian determinant and energy levels ε, and then determine the total number of bound states as well as the exact energy levels. Substitution of the obtained energy levels into the eigenfunction allows us to obtain the normalized eigenfunction, which can be illustrated easily in graphics. Finally, we notice that only the potential well U 2 exists a polynomial solution for some special potential well depths u.