Existence of ground state eigenvalues for the spin-boson model with critical infrared divergence and multiscale analysis

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:453

Article

Bach, Volker¹  Ballesteros, Miguel²  Konenberg, Martin³  Menrath, Lars¹ 

[1] Tech Univ Carolo Wilhelmina Braunschweig, Inst Anal & Algebra, Braunschweig, Germany

[2] Univ Nacl Autonoma Mexico, IIMAS, Mexico City, DF, Mexico

[3] Univ Stuttgart, Inst Anal Dynam & Modellierung, Stuttgart, Germany

关键词: Mathematical physics;    Multi-scale analysis;    Non-relativistic QED;    Quantum field theory;    Spectral theory;    Spin-boson;   

DOI  :  10.1016/j.jmaa.2017.03.075

来源: Elsevier

【 摘 要 】

A two-level atom coupled to the radiation field is studied. First principles in physics suggest that the coupling function, representing the interaction between the atom and the radiation field, behaves like vertical bar k vertical bar(-1/2), as the photon momentum k tends to zero. Previous results on non-existence of ground state eigenvalues suggest that in the most general case binding does not occur in the spin-boson model, i.e., the minimal energy of the atom-photon system is not an eigenvalue of the energy operator. Hasler and Herbst have shown [12], however, that under the additional hypothesis that the coupling function be off-diagonal - which is customary to assume - binding does indeed occur. In this paper an alternative proof of binding in case of off-diagonal coupling is given, i.e., it is proven that, if the coupling function is off-diagonal, the ground state energy of the spin-boson model is an eigenvalue of the Hamiltonian. We develop a multiscale method that can be applied in the situation we study, with the help of a key symmetry operator which we use to demonstrate that the most singular terms appearing in the multiscale analysis vanish. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free