Polynomial potentials and nilpotent groups

TL;DR

The paper develops a unified algebraic approach to quasi-exact solvability for one-dimensional Schrödinger operators with polynomial potentials of degree $2N-2$, built from irreducible representations of the nilpotent group $\mathcal{G}_N$ via the Hamiltonian $H_N = X_0^2 + X_N^2 + \alpha X_{N-1}$. By positing eigenfunctions of the form $\psi_E(x) = p(X_2) e^{-\int dx\,X_N}$ and expressing $X_N$ and $X_{N-1}$ through Casimirs, the authors derive an overdetermined system for the polynomial coefficients in $p$, yielding solvability conditions that fix $\alpha$ and constrain Casimirs. They provide explicit energy eigenvalues and eigenfunctions for quasi-exactly solvable sextic ($N=4$), octic ($N=5$), and decatic ($N=6$) potentials, including symmetrized odd/even variants and $E=0$ solutions for general $N$ and $M$, and discuss the electromagnetic-field interpretation via reducible representations. The work extends known results, offers new analytic solutions, and suggests broad applicability to solvable polynomial interactions and physically relevant field problems, while connecting to $sl(2,\mathbb{R})$-based algebraizations and beyond. The framework supplies a practical toolkit for constructing and testing quasi-exactly solvable models in quantum mechanics.

Abstract

This paper deals with the partial solution of the energy-eigenvalue problem for one-dimensional Schrödinger operators of the form $H_N=X_0^2+V_N$, where $V_N=X_N^2+αX_{N-1}$ 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 $α$ 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.

Figures (8)

• Figure 1: The sextic potential $(X_4^2+\alpha X_3)$ for $\alpha=-1$ (left) and $\alpha=-\frac{5}{3}$ (right), $\beta_1=6.$, $\beta_2=2.$, $\beta_3=-0.2$ and $\beta_4=\frac{\beta_2 \beta_3}{\beta_1}-\frac{\beta_2^3}{3 \beta_1^2}$ (i.e. $C_3=0$) along with the corresponding analytically calculable energy eigenvalues and eigenfunctions. Potential and wave functions are plotted as functions of $y=\arctan x$. The normalization of the wave function has been chosen such that $\int_{-\pi/2}^{\pi/2} dy\,{\Psi_M^\mathrm{sext}}^2(x(y))=1$.
• Figure 2: Same as in Fig. \ref{['fig:sext01']}, but for $\alpha=-\frac{7}{3}$ (left) and $\alpha=-3$ (right), corresponding to $M=2$ and $M=3$, respectively.
• Figure 3: Same as in Fig. \ref{['fig:sext01']}, but for $\alpha=-\frac{11}{3}$ (left) and $\alpha=-\frac{13}{3}$ (right), corresponding to $M=4$ and $M=5$, respectively. For better visability the wave functions have been normalized to $10$ rather than $1$.
• Figure 4: The symmetrized sextic potential for $\alpha=-\frac{7}{3}$, $\beta_4=0.5$ (left) and $\beta_4=-0.5$ (right), $\beta_1=16\beta_4^4$, $\beta_2=8\beta_4^3$ and $\beta_3=\frac{10}{3} \beta_4^2$ along with the corresponding analytically calculable energy eigenvalues and eigenfunctions of the parity even ground state and the parity odd excited state. Potential and wave functions are plotted as functions of $y=\arctan x$. The normalization of the wave function has been chosen such that $\int_{-\pi/2}^{\pi/2} dy\,{\Psi_2^{\mathrm{sext}\pm}}^2(x(y))=1$.
• Figure 5: The symmetrized octic potential for $\alpha=-2$, $\beta_2=2.$, $\beta_5=0.5$ and $\beta_1$, $\beta_2$, $\beta_4$ chosen according to Eq. (\ref{['eq:octM2even']}), upper sign (left), and Eq. (\ref{['eq:octM2odd']}) (right), respectively. Drawn are also the corresponding analytically calculable energy eigenvalues and eigenfunctions of even (left) and odd parity (right). Potential and wave functions are plotted as functions of $y=\arctan x$. The normalization of the wave function has been chosen such that $\int_{-\pi/2}^{\pi/2} dy\,{\Psi_2^{\mathrm{oct}\pm}}^2(x(y))=1$.