arXiv:1001.4313v3 [math-ph] 26 Mar 2010 Calculation of the Characteristic Functions of Anharmonic Oscillators Ulrich D. Jentschura Department of Physics, Missouri University of Science and Technology Rolla, Missouri, MO65409, USA Jean Zinn–Justin CEA, IRFU and Institut de Physique Th´ eorique, Centre de Saclay, F-91191 Gif-Sur-Yvette, France Abstract The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schr¨ odinger 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. MSC numbers 34E20, 81Q20, 81S99 Keywords Singular perturbations, turning point theory, WKB methods; Semiclassical techniques including WKB and Maslov methods; General quantum mechanics and problems of quantization 1 Introduction The cubic anharmonic oscillator, defined by the Hamiltonian, h 3 (g)= − 1 2 ∂ 2 ∂q 2 + 1 2 q 2 + √ gq 3 , (1) is a paradigmatic example of a quantum mechanical problem which gives rise to complex resonance energies. A particle initially trapped in the region q ≈ 0 may tunnel through the classically forbidden well and escape toward q → −∞. For positive coupling g> 0, the energies can be determined numerically by the method of complex scaling[2, 24]. One scales the coordinate as q → q e i θ for the cubic oscillator, which results in the Hamiltonian H c (θ) = e −2iθ  − 1 2 ∂ 2 ∂q 2 + 1 2 q 2 e 4iθ + √ gq 3 e 5iθ  . (2) 1