Summation of the eigenvalue perturbation series by multi-valued Pade approximants

Application to resonance problems and double wells

A. V. Sergeev

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

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 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.

Original languageEnglish
Article number030
Pages (from-to)4157-4162
Number of pages6
JournalJournal of Physics A: Mathematical and General
Volume28
Issue number14
DOIs
Publication statusPublished - 1995
Externally publishedYes

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Resonance Problem
Padé Approximants
Summation
Electron energy levels
Anharmonic Oscillator
eigenvalues
Eigenvalue
Perturbation
perturbation
Series
oscillators
Double-well Potential
Energy
Energy Levels
Energy Function
Degeneracy
Perturbation Theory
Continuation
energy
Branch

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

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