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

## TL;DR

This paper introduces a method to construct fully solvable quantum systems with customizable energy spectra using polynomial functions of harmonic oscillator Hamiltonians, revealing unique eigenfunction properties.

## Contribution

It presents a novel approach to design quantum systems with arbitrary discrete energy levels via polynomial transformations of harmonic oscillators.

## Key findings

- Energy eigenvalues can be freely chosen within the polynomial framework.
- Eigenfunctions exhibit non-monotonic node number behavior.
- Universal features of these polynomial-based quantum systems are identified.

## Abstract

We show that a polynomial H(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 such that the number of their nodes does not increase monotonically with increasing level number. Systems H have certain universal features, we study their basic behaviours.