# zbMATH — the first resource for mathematics

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 $$k$$.

##### MSC:
 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
##### Keywords:
limiting absorption principle; Fermi golden rule
Full Text: