  The energy levels of quantum systems are determined by quantization
conditions. For one-dimensional anharmonic oscillators, one can transform the
Schrodinger 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.
