Spectral theory of Pauli-Fierz operators. (English) Zbl 1034.81016

Pauli-Fierz Hamiltonians are self-adjoint operators that are used in quantum mechanics to describe the interaction of a small system with a Bose field. They are operators of the form \(H=H_{fr}+\lambda V\) on a Hilbert space \(\mathcal H\). This Hilbert space has a distinguished decomposition \({\mathcal H}={\mathcal H}^\nu \oplus {\mathcal H}^{\bar\nu}\). The “free operator” has the form \[ H_{fr} = \begin{pmatrix} H_{fr}^{\nu\nu} &0\\ 0 &H_{fr}^{\bar\nu\bar\nu}\end{pmatrix} \] and the perturbation has the form \[ V = \begin{pmatrix} 0 &V^{\nu\bar\nu}\\ V^{\bar\nu\nu} &V^{\bar\nu\bar\nu}. \end{pmatrix} \] The main results of the paper are the following.
a) Outside of an \(O(\lambda^2)\) neighborhood of \(\sigma(H^{\nu\nu})\) the spectrum of \(H\) is purely absolutely continuous and the Limiting Absorption Principle holds.
b) Let \(k\) be an isolated eigenvalue of \(H^{\nu\nu}\) and suppose the Fermi Golden Rule assumption for \(k\) holds. Then the spectrum of \(H\) is purely absolutely continuous in an \(O(\lambda^2)\) neighborhood of \(k\) and the Limiting Absorption Principle holds.
c) If the Fermi Golden Rule assumption fails, the description of the spectrum of \(H\) in a neighborhood of \(k+\lambda^2 m\) is obtained, where \(m\in\sigma_{disc}(w_k)\cap {\mathbb R}\), \(w_k=p_kw(z+i0)p_k\), \(w(z)=V^{\nu\bar\nu} (zI^{\bar\nu\bar\nu}- H_{fr}^{\bar\nu\bar \nu})^{-1}V^{\bar\nu\nu},\) and \(p_k\) is the projection of \(H_{fr}^{\nu\nu}\) onto \(k\).


81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47N50 Applications of operator theory in the physical sciences
81U05 \(2\)-body potential quantum scattering theory
Full Text: DOI


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