Semi-exact solutions to position-dependent mass Schrödinger problem with a class of hyperbolic potential V0tanh(ax)

Semiexact Solutions of the Razavy Potential

“…This is unlike previous study [22,24], in which the quasiexact wave functions and eigenvalues can be obtained by studying those two constraints. The present case is similar to our previous study [20,21], in which some constraint is violated. We have to choose other approach to study the eigenvalues as used in [20,21].…”

“…The present case is similar to our previous study [20,21], in which some constraint is violated. We have to choose other approach to study the eigenvalues as used in [20,21].…”

“…Generally speaking, the wave function requires ( ) → 0 when → ∞; i.e., → ∞. Unfortunately, the present study is unlike our previous study [20,21], in which → 1 when goes to infinity. The energy spectra can be calculated by series expansions through taking → 1.…”

“…In order to obtain some so-called exact solutions, the author has to take some constraints on the coefficients in the recurrence relations as shown in [18]. Inspired by recent study of the hyperbolic type potential well [20][21][22][23][24][25][26][27][28], in which we have found that their solutions can be exactly expressed by the confluent Heun functions [23], in this work we attempt to study the solutions of the Razavy potential. We shall find that the solutions can be written as the confluent Heun functions but their energy levels have to be calculated numerically since the energy term is involved within the parameter of the confluent Heun functions ( , , , , , ).…”

“…Even though the Heun functions have been studied well, its main topics are focused in the mathematical area. Only recent connections with the physical problems have been discovered; in particular the quantum systems for those hyperbolic type potential have been studied [20][21][22][23][24][25][26][27][28]. The terminology "semiexact" solutions used in [21] arise from the fact that the wave functions can be obtained analytically, but the eigenvalues cannot be written out explicitly.…”