### Abstract

We improve and generalize in several accounts the recent rigorous proof of convergence of delta expansion-order dependent mappings (variational perturbation expansion) for the energy eigenvalues of quartic anharmonic oscillator. For the single-well oscillator the uniformity of convergence in g∈[0, ∞] is proven. The convergence proof is extended also to complex values of g lying on a wide domain of the Riemann surface of E(g). Via the scaling relation à la Symanzik, this proves the convergence of delta expansion for the double well in the strong coupling regime (where the standard perturbation series is non Borel summable), as well as for the complex "energy eigenvalues" in certain metastable systems. Difficulties in extending the convergence proof to the cases of higher anharmonic oscillators are pointed out. Sufficient conditions for the convergence of delta expansion are summarized in the form of three general theorems, which should apply to a wide class of quantum mechanical and higher dimensional field theoretic systems.

Original language | English |
---|---|

Pages (from-to) | 109-145 |

Number of pages | 37 |

Journal | Annals of Physics |

Volume | 249 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 10 1996 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

*Annals of Physics*,

*249*(1), 109-145. https://doi.org/10.1006/aphy.1996.0066

**Improved convergence proof of the delta expansion and order dependent mappings.** / Guida, Riccardo; Konishi, Kenichi; Suzuki, Hiroshi.

Research output: Contribution to journal › Article

*Annals of Physics*, vol. 249, no. 1, pp. 109-145. https://doi.org/10.1006/aphy.1996.0066

}

TY - JOUR

T1 - Improved convergence proof of the delta expansion and order dependent mappings

AU - Guida, Riccardo

AU - Konishi, Kenichi

AU - Suzuki, Hiroshi

PY - 1996/7/10

Y1 - 1996/7/10

N2 - We improve and generalize in several accounts the recent rigorous proof of convergence of delta expansion-order dependent mappings (variational perturbation expansion) for the energy eigenvalues of quartic anharmonic oscillator. For the single-well oscillator the uniformity of convergence in g∈[0, ∞] is proven. The convergence proof is extended also to complex values of g lying on a wide domain of the Riemann surface of E(g). Via the scaling relation à la Symanzik, this proves the convergence of delta expansion for the double well in the strong coupling regime (where the standard perturbation series is non Borel summable), as well as for the complex "energy eigenvalues" in certain metastable systems. Difficulties in extending the convergence proof to the cases of higher anharmonic oscillators are pointed out. Sufficient conditions for the convergence of delta expansion are summarized in the form of three general theorems, which should apply to a wide class of quantum mechanical and higher dimensional field theoretic systems.

AB - We improve and generalize in several accounts the recent rigorous proof of convergence of delta expansion-order dependent mappings (variational perturbation expansion) for the energy eigenvalues of quartic anharmonic oscillator. For the single-well oscillator the uniformity of convergence in g∈[0, ∞] is proven. The convergence proof is extended also to complex values of g lying on a wide domain of the Riemann surface of E(g). Via the scaling relation à la Symanzik, this proves the convergence of delta expansion for the double well in the strong coupling regime (where the standard perturbation series is non Borel summable), as well as for the complex "energy eigenvalues" in certain metastable systems. Difficulties in extending the convergence proof to the cases of higher anharmonic oscillators are pointed out. Sufficient conditions for the convergence of delta expansion are summarized in the form of three general theorems, which should apply to a wide class of quantum mechanical and higher dimensional field theoretic systems.

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U2 - 10.1006/aphy.1996.0066

DO - 10.1006/aphy.1996.0066

M3 - Article

VL - 249

SP - 109

EP - 145

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

IS - 1

ER -