Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation

Abstract

Abstract.For a large class of quantum mechanical models of matter and radiation we develop an analytic perturbation theory for non-degenerate ground states. This theory is applicable, for example, to models of matter with static nuclei and non-relativistic electrons that are coupled to the UV-cutoff quantized radiation field in the dipole approximation. If the lowest point of the energy spectrum is a non-degenerate eigenvalue of the Hamiltonian, we show that this eigenvalue is an analytic function of the nuclear coordinates and of α 3/2 , α being the fine structure constant. A suitably chosen ground state vector depends analytically on α 3/2 and it is twice continuously differentiable with respect to the nuclear coordinates.

Ground State and Resonances in the Standard Model of the Non-Relativistic QED

“…In Appendix A we recall some properties of the Feshbach-Schur map and in Appendix B we prove the main result of Section VI. The results of both appendices are close to certain results from [4,34,29], but there are a few important differences. The main ones are that we have to deal with unbounded interactions and, more importantly, with momentum-anisotropic spaces.…”

“…As was mentioned in Section VII, the proof follows the lines of the proof of Theorem IV.3 of [29] (cf. Theorem 3.8 of [4] and Theorem 28 of [34]). It is similar to the proofs of related results of [10,11].…”

A New Asymptotic Perturbation Theory with Applications to Models of Massless Quantum Fields

“…(A.4) The function ω is continuous on R ν with lim |k|→∞ ω(k) = ∞ and there exist constants γ > 0 and C > 0 such that 12) where n 0 := max{n ≥ 0|E n < E 0 + ω 0 }. In particular, if the boson is massless, then σ(H 0 ) = [E 0 , ∞), which shows that all the eigenvalues of H 0 are embedded eigenvalues of H 0 .…”

“…For the latter, Bach, Fröhlich and Sigal [8,9]) have developed a method exploring a renormalization group idea combined with the Feshbach map. Recently analyticity in the coupling constant for the ground state energy 1 for some concrete models of massless quantum fields (in which the ground state energy of the unperturbed system under consideration is an embedded eigenvalue) has been proved [1,2,12,13]. These results are very nice, but, they may be model-dependent.…”

On the smooth Feshbach–Schur map

“…This generalization is needed, for example, in the analysis of resonances, and in the perturbation theory of the ground state of models of matter and quantized radiation [4,6]. In the course of modifying the proof of Theorem II.1 [3], we closely examined all of its parts.…”