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

TL;DR

This paper derives analytical and approximate solutions for a class of logarithmic potentials in a one-dimensional box, including energy spectra calculations, using transformations and hypergeometric functions.

Contribution

It introduces a new class of solvable logarithmic potentials in a confined space and provides methods for energy spectrum computation.

Findings

Analytic zero-energy solutions for the potential class.

Approximate solutions for non-zero energy with strong attraction.

Numerical scheme for energy spectrum calculation.

Abstract

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.