The energy levels for simplified anharmonic oscillators

Abstract

The energy levels for quantum mechanical oscillators with interaction imitatingx ^α (for integer α>2) are found by perturbative methods in finite number of dimensions. It is argued that in the limit of infinite dimensional space the coefficients in the expansion for the energy of theith level are growing with the perturbation ordern like$\frac{{(\frac{n}{2}\alpha + i - 1)!}}{{(i - 1)!(\frac{n}{2})!^2 }}\frac{n}{2}\alpha ^{1 - n}$. For the ground state (i=1) this reproduces estimates established for anharmonic oscillators.