Summation of the eigenvalue perturbation series by multi-valued Pade approximants: application to resonance problems and double wells

Abstract

Quadratic Pade approximants are used to obtain energy levels both for the anharmonic oscillator x2/2- lambda x4 and for the double well -x2/2+ lambda x4. In the first case, the complex-valued energy of the resonances is reproduced by summation of the real terms of the perturbation series. The second case is treated formally as an anharmonic oscillator with a purely imaginary frequency. We use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function. Then, we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants. In this way we can reconstruct the real energy by summation of the complex series. Such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory (around the minima of the double well potential) incorrect.