Calculation of the characteristic functions of anharmonic oscillators

• Published in: Applied numerical mathematics Vol. 60; no. 12; pp. 1332 - 1341
• Main Authors: Jentschura, Ulrich D., Zinn-Justin, Jean
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-12-2010; Elsevier
• Subjects: Anharmonicity; Derivatives; General quantum mechanics and problems of quantization; Mathematical analysis; Mathematical models; Oscillators; Physics; Quantization; Quantum Physics; Semiclassical techniques including WKB and Maslov methods; Shape; Singular perturbations; Turning point theory; Wave functions; WKB methods; Semiclassical techniques including WKB and Maslov methods; Singular perturbations; Turning point theory; General quantum mechanics and problems of quantization; WKB methods
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2010.03.015

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
+
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
m
=
3
.