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

We show that a polynomial ${\hat{H}}_{\left(N\right)}$ 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 $\hat{H}$ such that the number of their nodes does not increase monotonically with increasing level number. Systems $\hat{H}$ have certain 'universal' features, we study their basic behaviours.