SUMMATION OF THE EIGENVALUE PERTURBATION-SERIES BY MULTIVALUED PADE APPROXIMANTS - APPLICATION TO RESONANCE PROBLEMS AND DOUBLE WELLS

Quadratic Pade approximants are used to obtain energy levels both for the anharmonic oscillator x(2)/2-lambda x(4) and for the double well - x(2)/2 + lambda x(4). In the first case, the complex-valued energy of the resonances is reproduced by summation of the real terms of the per turbation series. The second case is treated formally as an anharmonic oscillator with a purely imaginary frequency. We use the expansion ar ound the central maximum of the potential to obtain a complex perturba tion series on the unphysical sheer, of the energy function. Then, we perform an analytical continuation of this solution to the neighbourin g physical sheet taking into account the supplementary branch of quadr atic approximants. In this way we can reconstruct the real energy by s ummation of the complex series. Such an unusual approach eliminates th e double degeneracy of states that makes ordinary perturbation theory (around the minima of the double-well potential) incorrect.