cea-02895309
https://cea.hal.science/cea-02895309
https://cea.hal.science/cea-02895309/document
https://cea.hal.science/cea-02895309/file/JJenbb.pdf
doi:10.1016/j.apnum.2010.03.015
[CEA] CEA - Commissariat à l'énergie atomique
[DSM-IRFU] IRFU
[CEA-DRF] Direction de Recherche Fondamentale
Calculation of the characteristic functions of anharmonic oscillators
D. Jentschura, Ulrich
Zinn-Justin, Jean
[PHYS.QPHY] Physics [physics]/Quantum Physics [quant-ph]
ART
Turning point theory
Singular perturbations
General quantum mechanics and problems of quantization
Semiclassical techniques including WKB and Maslov methods
WKB methods
turning point theory
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 .
2010-12
2020-07-09
en
Applied Numerical Mathematics
Elsevier