  We analyze quantitatively the accuracy of eigenfunction and eigenvalue
calculations in the frame work of WKB and instanton semiclassical methods. We
show that to estimate the accuracy it is enough to compare two linearly
independent (with the same quantum number) solutions to the Schr\"odinger
equations with the potential under study, and with the approximating piecewise
smooth potential. The main advantage of the approach is related to the
appropriate choice of the approximating potential, providing absolutely
convergent majorant series for the solutions. We test our method for a strongly
anharmonic one dimensional potential, but the basic ideas inspiring our work
and its results can be applied to a large variety of interesting chemical and
physical problems which are of relevance to various molecular systems.
