Calculation of the characteristic functions of anharmonic oscillators

• Autores: Ulrich D. Jentschura, Jean Zinn-Justin
• Localización: Applied numerical mathematics, ISSN-e 0168-9274, Vol. 60, Nº. 12, 2010, págs. 1332-1341
• Idioma: inglés
• DOI: 10.1016/j.apnum.2010.03.015
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• Resumen

  • 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 View the MathML source, 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.