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

A. V. Sergeev

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.

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