Scaling limit of a model of quantum electrodynamics

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.