Accuracy of the WKB approximation: The case of general quartic potential

We analyse the accuracy of the approximate WKB quantization for the case of general one-dimensional quartic potential. In particular, tee are interest ed in the validity of semiclassically predicted energy eigenvalues when app roaching the limit E --> infinity, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric doub le-well quartic potential. In the latter case, using the standard WKB quant ization an unnatural localization of eigenstates due to the negligence of t unneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order corrections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical appro ximation greatly by many orders of magnitude.