Solutions to position-dependent mass quantum mechanics for a new class of hyperbolic potentials

Abstract

We analytically solve the position-dependent mass (PDM) 1D Schrödinger equation for a new class of hyperbolic potentials $V_q^p(x) = -V_0\frac{\sinh ^px}{\cosh ^qx}, \, p= -2, 0, \dots q\,$Vqp(x)=−V0sinhpxcoshqx,p=−2,0,⋯q [see C. A. Downing, J. Math. Phys. 54, 072101 (2013)] among several hyperbolic single- and double-wells. For a solitonic mass distribution, \documentclass[12pt]{minimal}\begin{document}$m(x)=m_0\,\operatorname{sech}^2(x)$\end{document}m(x)=m0sech2(x), we obtain exact analytic solutions to the resulting differential equations. For several members of the class, the quantum mechanical problems map into confluent Heun differential equations. The PDM Poschl-Teller potential is considered and exactly solved as a particular case.