A generalized semiclassical expression for the eigenvalues of multiple well potentials

From the poles of a generalized semiclassical Green's function we derive expressions for the eigenvalues of I D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because- they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.