Polynomial potentials and nilpotent groups

This paper deals with the partial solution of the energy-eigenvalue problem for one-dimensional Schrödinger operators of the form $$H_N=X_0^2+V_N$$ H N = X 0 2 + V N , where $$V_N=X_N^2+\alpha X_{N-1}$$ V N = X N 2 + α X N - 1 is a polynomial potential of degree $$(2N-2)$$ ( 2 N - 2 ) and $$X_i$$ X i are the generators of an irreducible representation of a particular nilpotent group $${\mathcal{G}}_N$$ G N . Algebraization of the eigenvalue problem is achieved for eigenfunctions of the form $$\sum _{k=0}^M a_k X_2^k \exp (-\int dx\, X_N)$$ ∑ k = 0 M a k X 2 k exp ( - ∫ d x X N ) . It is shown that the overdetermined linear system of equations for the coefficients $$a_k$$ a k has a nontrivial solution, if the parameter $$\alpha$$ α and $$(N-3)$$ ( N - 3 ) Casimir invariants satisfy certain constraints. This general setting works for even $$N\ge 2$$ N ≥ 2 and can also be applied to odd $$N\ge 3$$ N ≥ 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 $$E=0$$ 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.