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

Abstract. Quadratic Padé approximants are used to obtain energy levels both for the anharmonic oscillator and for the double well . 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 complex perturbation series on the unphysical sheet of the energy function. Then, we perform analytic continuation of this solution to the neighbor 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 unusual approach eliminates double degeneracy of states that makes the ordinary perturbation theory (around the minima of the double well potential) to be incorrect.

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Calculate Rayleigh - Schrodinger perturbation series for the quartic, cubic, sextic, octic anharmonic oscillators and the Barbanis potential (two-dimensional anharmonic oscillator) using Mathematica programs.