HAL CCSD
Calculation of the characteristic functions of anharmonic oscillators
D. Jentschura, Ulrich
Zinn-Justin, Jean
Department of Physics [Rolla] ; Missouri University of Science and Technology (Missouri S&T) ; University of Missouri System-University of Missouri System
Institut de Recherches sur les lois Fondamentales de l'Univers (IRFU) ; Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay
International audience
ISSN: 0168-9274
EISSN: 0168-9274
Applied Numerical Mathematics
Elsevier
cea-02895309
https://cea.hal.science/cea-02895309
https://cea.hal.science/cea-02895309/document
https://cea.hal.science/cea-02895309/file/JJenbb.pdf
https://cea.hal.science/cea-02895309
Applied Numerical Mathematics, 2010, 60 (12), pp.1332-1341. ⟨10.1016/j.apnum.2010.03.015⟩
DOI: 10.1016/j.apnum.2010.03.015
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.apnum.2010.03.015
en
Turning point theory
Singular perturbations
General quantum mechanics and problems of quantization
Semiclassical techniques including WKB and Maslov methods
WKB methods
turning point theory
[PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph]
info:eu-repo/semantics/article
Journal articles
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
info:eu-repo/semantics/OpenAccess