Analytic properties of the perturbation series

Abstract

In quantum mechanical perturbation theory, one is interested in solving the eigenvalue problem of the perturbed Hamiltonian H(λ) = H₀ + λV₀, where H₀ is the exactly soluble unperturbed Hamiltonian and V = λV₀ is the perturbation. The strength parameter λ is a real parameter that can be continuously "dialed" from λ = 0 to λ = 1. For a given value of λ, the perturbed state ψ(λ) can be related to the unperturbed state ϕ by means of a perturbation operator L(λ), ψ(λ) = L(λ)ϕ. A very convenient normalization of ψ(λ) is to require that its projection into the subspace of the degenerate unperturbed states is normalized to unity; we shall refer to this scheme of normalization as the P-normalization. In the Schrödinger perturbation theory, the pertubed state is ultimately normalized to unity; we shall refer to this scheme as the unit normalization. Speisman worked out the perturbation series for L(λ) under unit normalization scheme. This operator approach to perturbation problems allowed him to discus convergence phenomenon in terms of norms of relevant operators, giving a lower bound for the convergence radius. In this paper, we propose that treating λ as a complex parameter allows us to study the exact radius of convergence through locations of singularities of the perturbation series. When we thus extend the domain of λ into the complex λ-plane, the perturbed Hamiltonian loses its Hermiticity, so that propeties like orthogonality and completeness of perturbed state are not automatically guaranteed. However, since H(λ) is hermitian along the real-line, we expect that certain aspects of the properties of the eigenstates and eigenvalues of hermitian operators to be retained. We shall therefore refer to this extended Hamiltonian as a quasi-hermitian operator.