TY - JOUR

T1 - Localization of the number of photons of ground states in nonrelativistic QED

AU - Hiroshima, Fumio

N1 - Funding Information:
I thank M. Griesemer for pointing out an error in the first manuscript. This work is in part supported by Grant-in-Aid 13740106 for Encouragement of Young Scientists from the Ministry of Education, Science, Sports, and Culture.
Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2003/5

Y1 - 2003/5

N2 - One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2(ℝ3) ⊗ F ≅ L2 (ℝ3; F), where F is the Boson Fock space over L2(ℝ3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to ∩k = 1∞D(1 ⊗ Nk), where N denotes the number operator of F. Moreover, it is shown that for almost every electron position variable x ∈ ℝ3 and for arbitrary k ≥ 0, ∥(1 ⊗ Nk/2ψg(x)∥F ≤ Dke-δ|x|m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular ψg ∈ ∩k = 1∞D(eβ|x|m+1 ⊗ Nk) for 0 < 0 < δ/2 is obtained.

AB - One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2(ℝ3) ⊗ F ≅ L2 (ℝ3; F), where F is the Boson Fock space over L2(ℝ3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to ∩k = 1∞D(1 ⊗ Nk), where N denotes the number operator of F. Moreover, it is shown that for almost every electron position variable x ∈ ℝ3 and for arbitrary k ≥ 0, ∥(1 ⊗ Nk/2ψg(x)∥F ≤ Dke-δ|x|m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular ψg ∈ ∩k = 1∞D(eβ|x|m+1 ⊗ Nk) for 0 < 0 < δ/2 is obtained.

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U2 - 10.1142/S0129055X03001667

DO - 10.1142/S0129055X03001667

M3 - Article

AN - SCOPUS:30244506465

VL - 15

SP - 271

EP - 312

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 3

ER -