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

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