The stationary KdV hierarchy and so(2,1) as a spectrum generating algebra

TL;DR

This paper explores the connection between high order Lie symmetries, the realization of so(2,1) as a spectrum generating algebra, and the stationary KdV hierarchy in one-dimensional quantum systems, revealing deep mathematical relationships.

Contribution

It identifies specific potential families with high order Lie symmetries and shows their relation to the stationary KdV hierarchy and so(2,1) algebra realizations.

Findings

Families F and F' are related to the stationary KdV hierarchy.

Potential V(x) can realize so(2,1) as a spectrum generating algebra.

High order Lie symmetry is connected to nonlinear differential equations.

Abstract

The family F of all potentials V(x) for which the Hamiltonian H in one space dimension possesses a high order Lie symmetry is determined. A sub-family F', which contains a class of potentials allowing a realization of so(2,1) as spectrum generating algebra of H through differential operators of finite order, is identified. Furthermore and surprisingly, the families F and F' are shown to be related to the stationary KdV hierarchy. Hence, the "harmless" Hamiltonian H connects different mathematical objects, high order Lie symmetry, realization of so(2,1)-spectrum generating algebra and families of nonlinear differential equations. We describe in a physical context the interplay between these objects.