### 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 language | English |
---|---|

Article number | 030 |

Pages (from-to) | 4157-4162 |

Number of pages | 6 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 28 |

Issue number | 14 |

DOIs | |

Publication status | Published - 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

**Summation of the eigenvalue perturbation series by multi-valued Pade approximants : Application to resonance problems and double wells.** / Sergeev, A. V.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 28, no. 14, 030, pp. 4157-4162. https://doi.org/10.1088/0305-4470/28/14/030

}

TY - JOUR

T1 - Summation of the eigenvalue perturbation series by multi-valued Pade approximants

T2 - Application to resonance problems and double wells

AU - Sergeev, A. V.

PY - 1995

Y1 - 1995

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21844499437&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21844499437&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/28/14/030

DO - 10.1088/0305-4470/28/14/030

M3 - Article

VL - 28

SP - 4157

EP - 4162

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 14

M1 - 030

ER -