Improved optimization of perturbation theory: Applications to the oscillator energy levels and Bose-Einstein condensate critical temperature

Improving perturbation theory via a variational optimization has generally produced in higher orders an embarrassingly large set of solutions, most of them unphysical (complex). We introduce an extension of the optimized perturbation method which leads to a drastic reduction of the number of acceptable solutions. The properties of this method are studied and it is then applied to the calculation of relevant quantities in different ${\ensuremath{\phi}}^{4}$ models, such as the anharmonic oscillator energy levels and the critical Bose-Einstein condensation temperature shift $\ensuremath{\Delta}{T}_{c}$ recently investigated by various authors. Our present estimates of $\ensuremath{\Delta}{T}_{c}$, incorporating the most recently available six and seven loop perturbative information, are in excellent agreement with all the available lattice numerical simulations. This represents a very substantial improvement over previous treatments.