Eigenvalues of λx^2m anharmonic oscillators

Abstract

The ground state as well as excited energy levels of the generalized anharmonic oscillator defined by the Hamiltonian H_m = - d²/dx²+x²+ λx²m, m = 2,3,..., have been calculated nonperturbatively using the Hill determinants. For the λx⁴ oscillator, the ground state eigenvalues, for various values of λ, have been compared with the Borel-Pade sum of the asymptotic perturbation series for the problem. The agreement is excellent. In addition, we present results for some excited states for m = 2 as well as the ground and the first even excited states for m = 3 and 4. The behaviour of all the energy levels with respect to the coupling parameter shows a qualitative similarity to the ground state of the λx⁴ oscillator. Thus the results are model independent, as is to be expected from the WKB approximation. Our results also satisfy the scaling property that ε_n^(m)(λ)/λ^1/(m+1) tend to a finite limit for large λ, and always lie within the variational bounds, where available.