%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