Isospectral mapping for quantum systems with energy point spectra to polynomial quantum harmonic oscillators

We show that a polynomial (H) over cap ((N)) of degree N of a harmonic oscillator hamiltonian allows us to devise a fully solvable continuous quantum system for which the first N discrete energy eigenvalues can be chosen at will. In general such a choice leads to a re-ordering of the associated energy eigenfunctions of (H) over cap such that the number of their nodes does not increase monotonically with increasing level number. Systems (H) over cap have certain 'universal' features, we study their basic behaviours. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

A polynomial of a harmonic oscillator hamiltonian of degree N can lead to a fully solvable continuous quantum system with customizable energy eigenvalues. The re-ordering of energy eigenfunctions due to this choice does not necessarily result in a monotonic increase in the number of nodes. These systems exhibit 'universal' features and their basic behaviors are studied.