Double wells: Perturbation series summable to the eigenvalues and directly computable approximations

Abstract

We give a rigorous proof of the analyticity of the eigenvalues of the double-well Schrodinger operators and of the associated resonances. We specialize the Rayleigh-Schrodinger perturbation theory to such problems, obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method. By an approximation we obtain new series expansions directly computable, still summable, which, in the case of the Herbst-Simon model, can be given in an explicit form.