An asymptotic analysis and its application to the nonrelativistic limit of the Pauli–Fierz and a spin-boson model

An abstract asymptotic theory of a family of self-adjoint operators {Hκ}κ>0 acting in the tensor product of two Hilbert spaces is presented and it is applied to the nonrelativistic limit of the Pauli–Fierz model in quantum electrodynamics and of a spin-boson model. It is proven that the resolvent of Hκ converges strongly as κ→∞ and the limit is a pseudoresolvent, which defines an "effective operator" of Hκ at κ≈∞. As corollaries of this result, some limit theorems for Hκ are obtained, including a theorem on spectral concentration. An asymptotic estimate of the infimum of the spectrum (the ground state energy) of Hκ is also given. The application of the abstract theory to the above models yields some new rigorous results for them.