Convergence of perturbation theory, an analytic study

Abstract

We show that the divergence of the perturbation series of energy is contingent upon loss of completeness of energy eigenstates of the pseudo-hermitian Hamiltonian at complex values of the strength parameter. We also show that extra, divergent type singularities may exist for the perturbation series of P-normalized states. These divergent singularities are also restricted to complex values of the strength parameter. The root cause for these singularity behaviors may be traced to the fact that the non-hermiticity of H(λ) at complex values of λ allows non-orthogonal eigenstates which may therefore either merge or contribute to the PVR subspace, as the case may be. An important implication of these results is that we have proven that lack of completeness of eigenstates of pseudo-hermitian operators occur only at branch points interior to the domain D for which the eigenvalues and eigenstates are defined.