### Abstract

This article gives a solution to an open problem in the paper by Arai [J. Math. Phys. 31, 2653 (1990)]. In it is presented an abstract asymptotic theory of families of unitary operators {script U(κ)}_{κ>0} and self-adjoint operators {H_{κ}}_{κ>0} acting in the tensor product of two Hilbert spaces. It is proven that H _{∈}
^{ren}(V,κ), which represents a scaled total Hamiltonian of a coupled system of a one electron atom and a quantized radiation field, with parameters 0≤∈≤1, κ>0, and the electron mass renormalized, is unitarily equivalent to an operator H̃ _{∈}
^{ren}(V,κ), which can be regarded as a decoupled Hamiltonian. Applying the abstract asymptotic theory and the unitary equivalence, it is proven that the resolvent of H_{∈} ^{ren}(V,κ) strongly converges as κ→∞ to an operator which defines an effective potential of the electron. The effective potential is compared with that obtained in the paper mentioned above.

Original language | English |
---|---|

Pages (from-to) | 4478-4516 |

Number of pages | 39 |

Journal | Journal of Mathematical Physics |

Volume | 34 |

Issue number | 10 |

Publication status | Published - 1993 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*34*(10), 4478-4516.

**Scaling limit of a model of quantum electrodynamics.** / Hiroshima, Fumio.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 34, no. 10, pp. 4478-4516.

}

TY - JOUR

T1 - Scaling limit of a model of quantum electrodynamics

AU - Hiroshima, Fumio

PY - 1993

Y1 - 1993

N2 - This article gives a solution to an open problem in the paper by Arai [J. Math. Phys. 31, 2653 (1990)]. In it is presented an abstract asymptotic theory of families of unitary operators {script U(κ)}κ>0 and self-adjoint operators {Hκ}κ>0 acting in the tensor product of two Hilbert spaces. It is proven that H ∈ ren(V,κ), which represents a scaled total Hamiltonian of a coupled system of a one electron atom and a quantized radiation field, with parameters 0≤∈≤1, κ>0, and the electron mass renormalized, is unitarily equivalent to an operator H̃ ∈ ren(V,κ), which can be regarded as a decoupled Hamiltonian. Applying the abstract asymptotic theory and the unitary equivalence, it is proven that the resolvent of H∈ ren(V,κ) strongly converges as κ→∞ to an operator which defines an effective potential of the electron. The effective potential is compared with that obtained in the paper mentioned above.

AB - This article gives a solution to an open problem in the paper by Arai [J. Math. Phys. 31, 2653 (1990)]. In it is presented an abstract asymptotic theory of families of unitary operators {script U(κ)}κ>0 and self-adjoint operators {Hκ}κ>0 acting in the tensor product of two Hilbert spaces. It is proven that H ∈ ren(V,κ), which represents a scaled total Hamiltonian of a coupled system of a one electron atom and a quantized radiation field, with parameters 0≤∈≤1, κ>0, and the electron mass renormalized, is unitarily equivalent to an operator H̃ ∈ ren(V,κ), which can be regarded as a decoupled Hamiltonian. Applying the abstract asymptotic theory and the unitary equivalence, it is proven that the resolvent of H∈ ren(V,κ) strongly converges as κ→∞ to an operator which defines an effective potential of the electron. The effective potential is compared with that obtained in the paper mentioned above.

UR - http://www.scopus.com/inward/record.url?scp=21344485386&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21344485386&partnerID=8YFLogxK

M3 - Article

VL - 34

SP - 4478

EP - 4516

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

ER -