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

作者：Arai Asao^*

来源：Annales Henri Poincare, 2014, 15(6): 1145-1170.

DOI：10.1007/s00023-013-0271-7

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(lambda) of the operator satisfying lim (lambda -%26gt; 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 -%26gt; 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 lambda as lambda -%26gt; 0.

• 出版日期2014-6