12-150 Asao Arai

A New Asymptotic Perturbation Theory with Applications to Models of Massless Quantum Fields (228K, Latex 2e) Dec 6, 12

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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 $\lambda o 0$.

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