Asao Arai
A New Asymptotic Perturbation Theory with Applications to
Models of Massless Quantum Fields
(228K, Latex 2e)
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 {\it not necessarily isolated}.
In this paper, we present a new asymptotic perturbation theory
for an eigenvalue $E(\lambda)$ of
the operator $H(\lambda):=H_0+\lambda H_{
I}$ ($\lambda\in \BBR\setminus\{0\}$)
satisfying $\lim_{\lambda o 0}E(\lambda)=E_0$. 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 $E(\lambda)$
up to an arbitrary finite order of $\lambda$ as $\lambda o 0$.
We apply the abstract results to a model of massless quantum fields, called the generalized spin-boson model
(A. Arai and M. Hirokawa, {\it J. Funct. Anal}. {f 151} (1997), 455--503) and show that
the ground state energy of the model has asymptotic expansions in the coupling constant $\lambda$
as $\lambda o 0$.