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Ground states and spectrum of quantum electrodynamics of nonrelativistic particles. (English) Zbl 0978.81022

Let \(\mathcal F\) be a boson Fock space and \({\mathcal H}=L^2({\mathbb R}^{dN})\otimes {\mathcal F}\). The Pauli-Fierz Hamiltonian \[ H = {1\over 2}\sum_{j=1}^N (P^j\otimes I-\varepsilon A(x^j))^2+V_{ex}\otimes I + I\otimes H_f \] in the Hilbeert space \(\mathcal H\) describes a system of \(N\)-nonrelativistic particles bound on external potential \(V_{ex}\) and coupled to a massless quantized radiation field \(A(x^j)\). Here \(H_f\) is the free Hamiltonian in \(\mathcal F\); \(P^j=(p_1^j,\dots,p_d^j)\), \(p^j_\mu=-i\nabla_{x^j_\mu}\); the external potential \(V_{ex}\) is the sum of a quadratic potential and a multiplication operator, which is infinitesimally small with respect to the Laplacian. The coupling constant \(\varepsilon\) is such that \(0 < \varepsilon < \varepsilon_0\) for some \(\varepsilon_0 > 0\). An ultraviolet cut-off is imposed on the quantized radiation field.
The existence of the ground states of \(H\) is established. It is shown that there exist asymptotic annihilation and creation operators and the absolutely continuous spectrum of \(H\) is \([G,\infty)\) for some \(G\in{\mathbb R}\).

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81V80 Quantum optics
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
81V45 Atomic physics
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