Here is my Hamiltonian: $H_{\alpha, \beta} (q,p) = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2q^2 + \alpha q + \frac{\beta}{2}(pq^2 + q^2p) + m \frac{\beta^2}{2}q^4$. How can I prove that $H_{\alpha, \beta}$ and $H_{\alpha, \beta‘}$ are unitarily equivalent and how I can compute the energy spectrum of the theory? It seems that the energy levels can be computed only perturbatively, because of the presence of term like $q$ or $q^4$, but in the text of the problem there no reference to perturbation theory.