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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.
