The Mourre Theory for Analytically Fibered Operators

We present here some results obtained with C. G\’erard about the Mourre theory for a class of operators which contains among other examples matrix valued differential operators with constant coefficients an periodic Schr\"odinger operators. We refer the reader to $[5, 4] $ for further details The framework is the one of a Hilbert space $\mathcal{H} $ equal to $L^{2}(M, \mu;\mathcal{H}’)=\int^{\oplus_{\mathcal{H}’d}}\mu(k)$ where $\mathcal{H}’ $ is a separable Hilbert space and $(M, \mu) $ is a a-finite measured space. In this framework, the fibered operators are the self-adjoint direct integrals of the form $H_{0} = \int^{\oplus}H\mathrm{o}(k)d\mu(k) $. The class of operators which we are interested in is characterized by the three conditions i) the space $M $ is a real-analytic manifold and $\mu $ is a $C^{\infty}- 1 $-density. ii) the resolvent $(H_{0}(k)+\dot{i})^{-1} $ is analytic with respect to $k $ and $H_{0}(k) $ has only discrete spectrum for $k\in M $. iii) the projection $p_{\mathrm{R}} $ : $\Sigma\ni(\lambda, k)-+\lambda\in \mathbb{R} $ is a proper map. The set $\Sigma $ mentionned above is the so called Bloch variety in the case of periodi