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Conference on "Variational and spectral methods in Quantum Field Theory"
Organizers: Volker Bach (Braunschweig), Maria J. Esteban (CNRS), Mathieu Lewin (CNRS), Eric Séré (Paris-Dauphine)


Miguel Ballesteros (Braunschweig, Germany): "Existence and Construction of Resonances for Atoms Coupled to the Quantized Radiation Field"
Monday 22 April 2013, 11:00 - 11:45
Amphi Hermite


The theory of quantum mechanics asserts that the energy of an atom can be quantized. The possible energies are the eigenvalues of the Schrödinger equation. The lowest eigenvalue is called the ground state energy and the other eigenvalues are the excited energies. The Schrödinger equation predicts that if an atom is in an excited state at some time then it remains in that state forever. Experiments show, however, that the atom does not remain in an excited state forever, but decays to a lower energy state. During this decay, the atom emits photons whose energy is given by the difference between the initial and the final energy, in accordance with the Bohr's frequency condition. The process described above can be expressed on a technical level as follows: the atom remains in a certain state for a short period and then decays to a lower energy state. Thus the excited states are not eigenvalues of a certain Hamiltonian but turn into emph{resonances}. These resonances appear when the photon field is introduced into the picture.

We analyze the Pauli-Fierz model, which represents a non-relativistic atom coupled to a (quantized) photon field. We prove that the excited eigenvalues of the atom give rise to resonances, once the photon field is introduced, and that the energies of the resonance-producing photons are given by Bohr's frequency condition, up to second order in the coupling constant. We do not assume that there is an infrared regularization but we require an ultraviolet cutoff. We review Sigal's recent construction of resonances based on renormalization group analysis and present a novel alternative construction based on ``Pizzo's Method''.

This is a joint work with Volker Bach and Alessandro Pizzo.