https://cea.hal.science/cea-02895309D. Jentschura, UlrichUlrichD. JentschuraDepartment of Physics [Rolla] - Missouri S&T - Missouri University of Science and Technology - University of Missouri SystemZinn-Justin, JeanJeanZinn-JustinIRFU - Institut de Recherches sur les lois Fondamentales de l'Univers - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - Université Paris-SaclayCalculation of the characteristic functions of anharmonic oscillatorsHAL CCSD2010Turning point theorySingular perturbationsGeneral quantum mechanics and problems of quantizationSemiclassical techniques including WKB and Maslov methodsWKB methodsturning point theory[PHYS.QPHY] Physics [physics]/Quantum Physics [quant-ph]Savelli, Bruno2020-07-09 16:06:242021-12-13 09:15:482020-07-09 16:07:03enJournal articleshttps://cea.hal.science/cea-02895309/document10.1016/j.apnum.2010.03.015application/pdf1The 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 .