A New Asymptotic Perturbation Theory with Applications to Models of Massless Quantum Fields

Abstract

Let H 0 and H I be a self-adjoint and a symmetric operator on a complex Hilbert space, respectively, and suppose that H 0 is bounded below and the infimum E 0 of the spectrum of H 0 is a simple eigenvalue of H 0 which is not necessarily isolated. In this paper, we present a new asymptotic perturbation theory for an eigenvalue E(λ) of the operator $${H(\lambda)\,:=\,H_0 + \lambda H_{I}\,(\lambda \in \mathbb{R} \setminus \{0\})}$$ satisfying lim λ → 0 E(λ) = E 0. The point of the theory is in that it covers also the case where E 0 is a non-isolated eigenvalue of H 0. Under a suitable set of assumptions, we derive an asymptotic expansion of E(λ) up to an arbitrary finite order of λ as λ → 0. We apply the abstract results to a model of massless quantum fields, called the generalized spin-boson model (Arai and Hirokawa in J Funct Anal 151:455–503, 1997) and show that the ground-state energy of the model has asymptotic expansions in the coupling constant λ as λ → 0.