2024/12/15 by W. Schweiger, Schweiger, W., W. H. Klink +1
Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2412.11157
openalex publication_date 2024/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper deals with the partial solution of the energy-eigenvalue problem for one-dimensional Schrödinger operators of the form HN=X02+VN, where VN=XN2+αXN-1 is a polynomial potential of degree (2N-2) and Xi are the generators of an irreducible representation of a particular nilpotent group GN. Algebraization of the eigenvalue problem is achieved for eigenfunctions of the form ∑k=0M ak X2k exp(-∫ dx XN). It is shown that the overdetermined linear system of equations for the coefficients ak has a nontrivial solution, if the parameter α and (N-3) Casimir invariants satisfy certain constraints. This general setting works for even N≥ 2 and can also be applied to odd 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 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.