Accurate approximate analytical solutions for a double-well potential

We employ a series of transformations, nonlinear as well as linear, to the Hamiltonian operator which describes a generic symmetric double-well potential. The linear transformations exploit some properties of the Lie algebras SO(3) and SO(2, 1), and lead to a convenient decomposition of the operator for applications of Rayleigh–Schrödinger perturbation theory (RSPT). Since conventional RSPT is not universally effective, we provide an alternative novel procedure which leads to compact analytic wavefunctions as well as accurate eigenvalues.