Modified Riccati approach to partially solvable quantum Hamiltonians. III. Related families of Pöschl-Teller potentials.

Abstract

Within the modified Riccati approach developed elsewhere, we construct quasiexactly solvable potentials related to the P\"oschl-Teller and modified P\"oschl-Teller potentials. This is achieved by considering closed-form wave functions whose logarithmic derivatives are rational in the mapping u(x)=cosh(x) or u(x)=cos(x). With the hyperbolic mapping we construct an interesting four-parameter quasiexactly solvable confining potential with one, two, or three minima upon the value of its coupling constants. We give explicit expressions for the bound-state eigenfunctions and corresponding energy levels for some particular cases. The analytic continuation x\ensuremath{\rightarrow}ix transforms the previous potential into a periodic confining well, or a Kronig-Penney-like potential with an interesting structure of minima in each cell and a finite number of gaps. The energy levels in the last case are some of the band edges of the spectrum. In any case a finite but arbitrarily large number of energy levels and corresponding eigenfunctions are determined from the diagonalization of a finite matrix.