Perturbative technique as an alternative to the WKB method applied to the double-well potential

Abstract
We give an explicit and complete perturbation theoretical analysis of the solutions and the eigenvalues of the Schrödinger equation for the double-well potential. In particular we demonstrate the matching of various branches of the solutions over the entire range of the independent variable, and we calculate the splitting of eigenvalues due to the finite height of the central hump of the potential.