New solvable quantum mechanical potentials by iteration of the free V=0 potential

TL;DR

This paper introduces a method to generate a large family of solvable quantum potentials starting from the free potential, including new singular potentials with unique properties, linked to rational solutions of the KdV equation.

Contribution

It systematically exploits the Darboux method to produce new solvable potentials from the free case, expanding known families and discovering novel singular potentials.

Findings

Explicitly finds and solves new singular potentials

Identifies potentials with E=0 bound states and constant phase shift

Relates potentials to rational solutions of the KdV family

Abstract

A huge family of solvable potentials can be generated by systematically exploiting the factorization (Darboux) method. Starting from the free case, a large class of the known solvable families is thus reproduced, together with new ones. We explicitly find and solve several new singular potentials obtained by iteration from the V=0 case; some of them have an E=0 bound state and constant phase shift without being explicitly scale invariant. The new potentials are rational functions, and can be related to rational solutions of the KdV family.