### Abstract

The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrödinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr-Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is Bm(E,g)=n+12, where B is a characteristic function of the anharmonic oscillator of degree m, E is the resonance energy, and g is the coupling constant. A recursive scheme can be devised which facilitates the evaluation of higher-order Wentzel-Kramers-Brioullin (WKB) approximants. The WKB expansion of the logarithmic derivative of the wave function has a cut in the tunneling region. The contour integral about the tunneling region yields the instanton action plus corrections, summarized in a second characteristic function Am(E,g). The evaluation of Am(E,g) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m=3.

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

Pages (from-to) | 1332-1341 |

Number of pages | 10 |

Journal | Applied Numerical Mathematics |

Volume | 60 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2010 |

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### Keywords

- General quantum mechanics and problems of quantization
- Semiclassical techniques including WKB and Maslov methods
- Singular perturbations
- Turning point theory
- WKB methods

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

*Applied Numerical Mathematics*,

*60*(12), 1332-1341. https://doi.org/10.1016/j.apnum.2010.03.015

**Calculation of the characteristic functions of anharmonic oscillators.** / Jentschura, U.; Zinn-Justin, Jean.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 60, no. 12, pp. 1332-1341. https://doi.org/10.1016/j.apnum.2010.03.015

}

TY - JOUR

T1 - Calculation of the characteristic functions of anharmonic oscillators

AU - Jentschura, U.

AU - Zinn-Justin, Jean

PY - 2010/12

Y1 - 2010/12

N2 - The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrödinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr-Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is Bm(E,g)=n+12, where B is a characteristic function of the anharmonic oscillator of degree m, E is the resonance energy, and g is the coupling constant. A recursive scheme can be devised which facilitates the evaluation of higher-order Wentzel-Kramers-Brioullin (WKB) approximants. The WKB expansion of the logarithmic derivative of the wave function has a cut in the tunneling region. The contour integral about the tunneling region yields the instanton action plus corrections, summarized in a second characteristic function Am(E,g). The evaluation of Am(E,g) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m=3.

AB - The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrödinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr-Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is Bm(E,g)=n+12, where B is a characteristic function of the anharmonic oscillator of degree m, E is the resonance energy, and g is the coupling constant. A recursive scheme can be devised which facilitates the evaluation of higher-order Wentzel-Kramers-Brioullin (WKB) approximants. The WKB expansion of the logarithmic derivative of the wave function has a cut in the tunneling region. The contour integral about the tunneling region yields the instanton action plus corrections, summarized in a second characteristic function Am(E,g). The evaluation of Am(E,g) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m=3.

KW - General quantum mechanics and problems of quantization

KW - Semiclassical techniques including WKB and Maslov methods

KW - Singular perturbations

KW - Turning point theory

KW - WKB methods

UR - http://www.scopus.com/inward/record.url?scp=77958022440&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77958022440&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2010.03.015

DO - 10.1016/j.apnum.2010.03.015

M3 - Article

AN - SCOPUS:77958022440

VL - 60

SP - 1332

EP - 1341

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 12

ER -