Eigenvalues of λx2m anharmonic oscillators

The ground state as well as excited energy levels of the generalized anharmonic oscillator defined by the Hamiltonian H_m = − d²/dx²+x²+ λx^2m, 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‐Padé 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 ${\epsilon }_n^{\mathrm{(m)}}{\mathrm{(\lambda )/\lambda }}^{\text{1/(m+1)}}$ tend to a finite limit for large λ, and always lie within the variational bounds, where available.