Calculation of the characteristic functions of anharmonic oscillators

Ulrich D. Jentschura, Jean Zinn-Justin

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3 Citations (Scopus)


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 languageEnglish
Pages (from-to)1332-1341
Number of pages10
JournalApplied Numerical Mathematics
Issue number12
Publication statusPublished - Dec 1 2010



  • 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

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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