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# Calculation of the characteristic functions of anharmonic oscillators

Abstract : 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 .
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Ulrich D. Jentschura, Jean Zinn-Justin. Calculation of the characteristic functions of anharmonic oscillators. Applied Numerical Mathematics, Elsevier, 2010, 60 (12), pp.1332-1341. ⟨10.1016/j.apnum.2010.03.015⟩. ⟨cea-02895309⟩

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