Non-polynomial extensions of solvable potentials a la Abraham-Moses

TL;DR

This paper derives explicit seed solutions for Abraham-Moses transformations, enabling non-polynomial extensions of solvable quantum potentials, expanding the toolkit for potential modifications beyond Darboux methods.

Contribution

It provides explicit forms of seed solutions for Abraham-Moses transformations applicable to various solvable potentials, facilitating non-polynomial potential extensions.

Findings

Explicit seed solutions for Abraham-Moses transformations derived.

Non-polynomial extensions of solvable potentials constructed.

Applicable to potentials like radial oscillator and Pöschl-Teller.

Abstract

Abraham-Moses transformations, besides Darboux transformations, are well-known procedures to generate extensions of solvable potentials in one-dimensional quantum mechanics. Here we present the explicit forms of infinitely many seed solutions for adding eigenstates at arbitrary real energy through the Abraham-Moses transformations for typical solvable potentials, e.g. the radial oscillator, the Darboux-P\"oschl-Teller and some others. These seed solutions are simple generalisations of the virtual state wavefunctions, which are obtained from the eigenfunctions by discrete symmetries of the potentials. The virtual state wavefunctions have been an essential ingredient for constructing multi-indexed Laguerre and Jacobi polynomials through multiple Darboux-Crum transformations. In contrast to the Darboux transformations, the virtual state wavefunctions generate non-polynomial extensions of solvable potentials through the Abraham-Moses transformations.