Nilpotent Groups and Anharmonic Oscillators

Abstract

Representations of nilpotent groups relate different quantum systems having polynomial interactions. In this paper the nilpotent group associated with the quartic anharmonic oscillator is analyzed in detail and the relationship between the quartic anharmonic oscillator Hamiltonian and irreducible representations of Lie algebra elements of the nilpotent group is given. Scaling operators are used to partially determine the functional form of the eigenvalues. The Hamiltonian for a particle in a nonconstant magnetic field, as well as a “heat” Hamiltonian, are shown to be related to reducible representations of the nilpotent group. Generalizations to other nilpotent groups for which there are scaling operators are also given.