The anharmonic oscillator and its dependence on space dimensions

The perturbation series (PS) in powers of g for the D-dimensional gr4 anharmonic oscillator is considered. High-order terms of the ground-state-energy E(g,D) expansion are explicitly calculated with the help of recurrence relations among integers. The rate of approach of the PS coefficients to the asymptotic value as a function of D is investigated. For the case of D=0 explicit solutions are obtained and analytic properties of E(g,0) are briefly discussed. For D>>1 the first terms in the expansion of E(g,D) in powers of D-1 are found. (Auth.)