A class of singular logarithmic potentials in a box with variety of skin thickness and wall interaction

We obtain an analytic solution for a three-parameter class of logarithmic potentials at zero energy. The potential terms are products of the inverse square and the inverse log to powers 2, 1 and 0. The configuration space is the one-dimensional box. Using point canonical transformation, we simplify the solution by mapping the problem into the oscillator problem. We also obtain an approximate analytic solution for non-zero energy when there is strong attraction to one side of the box. The wavefunction is written in terms of the confluent hypergeometric function. We also present a numerical scheme to calculate the energy spectrum for a general configuration and to any desired accuracy.Comment: 9 pages, 5 figures, 3 table

Mapping Schr\"odinger equation into a Heun-type and identifying the corresponding potential function, energy and wavefunction

We transform the Schr\"odinger wave equation to a nine-parameter Heun-type differential equation. Using our solutions of the latter in [J. Math. Phys. 59 (2018) 113507], we are able to identify the associated potential function, energy parameter, and write the corresponding wave function. Some of the solutions obtained correspond to new integrable quantum systems.Comment: This revised version corrects a mistake in Eq. (7

Reply to 'Comment on "Relativistic extension of shape-invariant potentials"'

We concur with de Castro's observation that the gauge considerations of our approach are not valid. Nevertheless, except for an error that will be corrected, all of our findings are accurate independent of those considerations