Accuracy of Semiclassics: Comparative Analysis of WKB and Instanton Approaches

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.