Quantum Mathematical Physics by Felix Finster Johannes Kleiner Christian Röken & Jürgen Tolksdorf

Author:Felix Finster, Johannes Kleiner, Christian Röken & Jürgen Tolksdorf
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Radiation,

Scattering theory.

The headings above have the following translations into mathematical terms:

Stability Existence of the ground state,

Radiation

Instability and decay of the excited states of particle systems,

Scattering Asymptotic behaviour of evolution at large times.

Remarkable progress was made in the last 10 or so years in constructing the rigorous theory of the stability, radiation and scattering.

Three powerful and unifying techniques emerged as the result of this development: theory of resonances, giving a clear-cut mathematical description of processes of emission and absorption of the electro-magnetic radiation, the spectral renormalization group, giving the perturbation theory for bound states and resonances and the method of propagation observables and propagation inequalities underpinning the scattering theory.

In this brief review, we summarize progress on the first two items and give more details on the last one. In considering the long-time dynamics, we concentrate on energies below the ionization threshold (Rayleigh scattering). The review is based on the papers [35, 36]. We also borrow some general material from our previous review [73], to which we refer the reader for more details and references of the problems of stability and radiation (see also the review [4] and the book [79]).

We formulate the key problems related the long-time dynamics which is the area of concentration of this review. The Hamiltonian H of a non-relativistic particle system coupled to the quantized electromagnetic or phonon field is self-adjoint and generates the dynamics through the Schrödinger equation,

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