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# Autour du groupe de renormalisation, et ses applications en électrodynamique quantique non relativiste I.

Mardi, 17 Avril, 2007 - 12:00
Prénom de l'orateur :
Volker
Nom de l'orateur :
BACH
Résumé :

The smooth Feshbach map (SFM) is a generalization of a
well-known projection method (Feshbach projection method,
Lyapunov-Schmidt Reduction, Grushin Problem, Schur complement) for the
computation of spectra of operators in mathematics and physics. In
its simplest form it says that if $mathcal{H}$ is a Hilbert space, $P = P^2 = P^*$ is an orthogonal projection on $mathcal{H}$ and $H$ an
operator which is invertible on $P^perp mathcal{H}$, then $0$ is an
eigenvalue of $H$ iff $0$ is an eigenvalue of
$$F_P(H) := P H P - P H P^perp (P^perp H P^perp)^{-1} P^perp H P.$$
Here, $F_P(H)$
acts on the subspace $P mathcal{H} subset mathcal{H}$ and is,
hence, potentially easier to analyze. The above equivalence is refered
to as emph{isospectrality} (of the map $H mapsto F_P(H)$).
Iterating $F_P$ with a nested sequence $P_0 := 1 geq P_1 geq P_2 cdots$ of projections leads to a sequence $( H_0 := H, H_1 := F_{P_1}(H), H_2 := F_{P_2}(F_{P_1}(H)), ldots)$ of isospectral
operators acting on ever smaller subspaces of $mathcal{H}$.

For certain interacting models in nonrelativistic quantum
electrodynamics (QED), it is possible to show by renormalization group
methods that this sequence converges (in a suitable sense) to a
limiting operator $H_infty$ whose spectral analysis is trivial
because it represents a noninteracting model. Thanks to
isospectrality, one can go backwards in the iteration and obtain
information about $H = H_0$ from it. In particular, this allows to
prove the existence of a ground state eigenvector of the model and to
expand it and the corresponding eigenvalue (the ground state energy)
in a convergent series. Moreover, the control is explicit enough to
expand a further quantity, the renormalized electron mass, in convergent
series.

The results presented in these lectures were obtained in joint
collaboration with Thomas Chen, Jürg Fröhlich, and Israel Michael
Sigal.

Institution de l'orateur :
Université de Mayence
Thème de recherche :
Physique mathématique
Salle :
1 tour Irma