### Abstract

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 L^{2}(ℝ^{3}) ⊗ F ≅ L^{2} (ℝ^{3}; F), where F is the Boson Fock space over L^{2}(ℝ^{3} × {1, 2}). It is shown that the ground state, ψ_{g}, of H belongs to ∩_{k = 1} ^{∞}D(1 ⊗ N^{k}), where N denotes the number operator of F. Moreover, it is shown that for almost every electron position variable x ∈ ℝ^{3} and for arbitrary k ≥ 0, ∥(1 ⊗ N^{k/2}ψ_{g}(x)∥_{F} ≤ D_{k}e^{-δ|x|m+1} with some constants m ≥ 0, D_{k} > 0, and δ > 0 independent of k. In particular ψ_{g} ∈ ∩_{k = 1} ^{∞}D(e^{β|x|m+1} ⊗ N^{k}) for 0 < 0 < δ/2 is obtained.

Original language | English |
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Pages (from-to) | 271-312 |

Number of pages | 42 |

Journal | Reviews in Mathematical Physics |

Volume | 15 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2003 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Localization of the number of photons of ground states in nonrelativistic QED.** / Hiroshima, Fumio.

Research output: Contribution to journal › Article

*Reviews in Mathematical Physics*, vol. 15, no. 3, pp. 271-312. https://doi.org/10.1142/S0129055X03001667

}

TY - JOUR

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

AU - Hiroshima, Fumio

PY - 2003/5/1

Y1 - 2003/5/1

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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UR - http://www.scopus.com/inward/citedby.url?scp=30244506465&partnerID=8YFLogxK

U2 - 10.1142/S0129055X03001667

DO - 10.1142/S0129055X03001667

M3 - Article

VL - 15

SP - 271

EP - 312

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 3

ER -