TY - JOUR

T1 - Remarks on the ground state energy of the spin-boson model. An application of the Wigner-Weisskopf model

AU - Hirokawa, Masao

N1 - Funding Information:
H. Spohn proposed the problem of expressing ESB(0) independently of the existence of its ground state to me when I held discussions on [25] with him, though I had assumed the existence in [25]. So, this is the beginning of the problem I dealt with in this paper. I wish to thank him for giving me the beginning of the problem. I argued the problem about ESB(0) given by the limit (4.2) of the explicit expression for ESB(ν) in [25] with V. Bach and A. Elgart when I visited Technische Universität Berlin during September 8{10, ’98. Then the above problem for (4.1){(4.4) on the survival of µ arose. I wish to thank them for arrangements of my visiting Technische Universität Berlin and the hospitality. I am indebted to A. Arai for useful discussions which proofs in this paper were based on. I thank H. Spohn and F. Hiroshima for their hospitality at Technische Universität München during April 15{22, ’99, and discussing Spohn’s unpublished results. I wish to express H. Spohn, R. A. Minlos, H. Ezawa, K. Watanabe, K. Yasue, M. Jibu, F. Hiroshima, and the referee for valuable advice. I wish to thank J. Derezinski for discussing several aspects about the generalized spin-boson model at the summer school \Schrödinger Operators and Related Topics," Shonan Village Center, July 5{9, ’99, and also C. G erard for telling me how to get his recent result which broke through a wall in Theorem 2.2(c). My research is supported by the Grant-In-Aid No. 11740109 for Encouragement of Young Scientists from Japan Society for the Promotion of Science.

PY - 2001/2

Y1 - 2001/2

N2 - For the ground state energy of the spin-boson (SB) model, we give a new upper bound in the case with infrared singularity condition (i.e. without infrared cutoff), and a new lower bound in the case of massless bosons with infrared regularity condition. We first investigate spectral properties of the Wigner-Weisskopf (WW) model, and apply them to SB model to achieve our purpose. Then, as an extra result of the spectral analysis for WW model, we show that a non-perturbative ground state appears, and its ground state energy is so low that we cannot conjecture it by using the regular perturbation theory.

AB - For the ground state energy of the spin-boson (SB) model, we give a new upper bound in the case with infrared singularity condition (i.e. without infrared cutoff), and a new lower bound in the case of massless bosons with infrared regularity condition. We first investigate spectral properties of the Wigner-Weisskopf (WW) model, and apply them to SB model to achieve our purpose. Then, as an extra result of the spectral analysis for WW model, we show that a non-perturbative ground state appears, and its ground state energy is so low that we cannot conjecture it by using the regular perturbation theory.

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U2 - 10.1142/S0129055X01000727

DO - 10.1142/S0129055X01000727

M3 - Article

AN - SCOPUS:0035585522

SN - 0129-055X

VL - 13

SP - 221

EP - 251

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

IS - 2

ER -