### 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 |
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Pages (from-to) | 4478-4516 |

Number of pages | 39 |

Journal | Journal of Mathematical Physics |

Volume | 34 |

Issue number | 10 |

DOIs | |

Publication status | Published - Jan 1 1993 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Mathematical Physics*,

*34*(10), 4478-4516. https://doi.org/10.1063/1.530353