%0 Journal Article
%T Calculation of the characteristic functions of anharmonic oscillators
%+ Department of Physics [Rolla]
%+ Institut de Recherches sur les lois Fondamentales de l'Univers (IRFU)
%A D. Jentschura, Ulrich
%A Zinn-Justin, Jean
%< avec comité de lecture
%@ 0168-9274
%J Applied Numerical Mathematics
%I Elsevier
%V 60
%N 12
%P 1332-1341
%8 2010-12
%D 2010
%R 10.1016/j.apnum.2010.03.015
%K Turning point theory
%K Singular perturbations
%K General quantum mechanics and problems of quantization
%K Semiclassical techniques including WKB and Maslov methods
%K WKB methods
%K turning point theory
%Z Physics [physics]/Quantum Physics [quant-ph]Journal articles
%X 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 $B_m$($E,g)=n+$ $\frac{1}{2}$, 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 $A_m$($E,g$). The evaluation of $A_m$($E,g$) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree .
%G English
%2 https://cea.hal.science/cea-02895309/document
%2 https://cea.hal.science/cea-02895309/file/JJenbb.pdf
%L cea-02895309
%U https://cea.hal.science/cea-02895309
%~ CEA
%~ DSM-IRFU
%~ CEA-DRF