Polynomial potential and nilpotent groups

Dr. Wolfgang Schweiger

Abstract: The talk deals with the partial solution of the energy eigenvalue problem for one-dimensional Schroedinger operators with polynomial potentials. It is shown that algebraization of the problem is possible when the Schroedinger operator can be expressed in terms of the generators of certain nilpotent groups. These groups generalize the Heisenberg group that underlies the solvability of the harmonic oscillator. Demanding that the potential gives rise to eigenfunctions of polynomial form (times an exponential factor) puts constraints on the Casimir invariants and thus on the potential parameters. Since these constraints differ, in general, for different eigenfunctions, the polynomial ansatz for the eigenfunctions will thus provide only a restricted number of eigenvalues and eigenfunctions. In the literature, spectral problems for which only a partial solution is known in analytical form are usually called "quasi-exactly solvable". We will show how the harmonic, quartic, sextic, octic and the decatic oscillator can be treated in a unified way within our approach and present corresponding examples. Other approaches to quasi-exactly solvable problems are also shortly discussed.

Literature: William H. Klink and Wolfgang Schweiger, Polynomial solutions of generalized quartic anharmonic oscillators, Eur.Phys.J.Plus 138 (2023) 656. Wolfgang Schweiger and William H. Klink, Polynomial potentials and nilpotent groups, Eur.Phys.J.Plus 140 (2025) 559.

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