Polynomial potentials and nilpotent groups

TL;DR

This paper presents a unified algebraic approach to solving the energy eigenvalue problem for a class of polynomial potentials associated with nilpotent groups, extending quasi-exact solvability to higher-degree potentials.

Contribution

It introduces a general method for quasi-exactly solving Schrödinger equations with polynomial potentials linked to nilpotent groups, including explicit solutions for sextic, octic, and decatic cases.

Findings

Provides explicit energy eigenvalues and eigenfunctions for specific polynomial potentials.

Establishes conditions for nontrivial solutions based on parameters and Casimir invariants.

Extends quasi-exact solvability to higher-degree polynomial potentials.

Abstract

This paper deals with the partial solution of the energy-eigenvalue problem for one-dimensional Schr\"odinger operators of the form $H_N=X_0^2+V_N$, where $V_N=X_N^2+\alpha X_{N-1}$ is a polynomial potential of degree $(2N-2)$ and $X_i$ are the generators of an irreducible representation of a particular nilpotent group $\mathcal{G}_N$. Algebraization of the eigenvalue problem is achieved for eigenfunctions of the form $\sum_{k=0}^M a_k X_2^k \exp(-\int dx\, X_N)$. It is shown that the overdetermined linear system of equations for the coefficients $a_k$ has a nontrivial solution, if the parameter $\alpha$ and $(N-3)$ Casimir invariants satisfy certain constraints. This general setting works for even $N\geq 2$ and can also be applied to odd $N\geq 3$, if the potential is symmetrized by considering it as function of $|x|$ rather than $x$. It provides a unified approach to quasi-exactly solvable polynomial interactions, including the harmonic oscillator, and extends corresponding results known from the literature. Explicit expressions for energy eigenvalues and eigenfunctions are given for the quasi-exactly solvable sextic, octic and decatic potentials. The case of $E=0$ solutions for general $N$ and $M$ is also discussed. As physical application, the movement of a charged particle in an electromagnetic field of pertinent polynomial form is shortly sketched.