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.