The adiabatic limit of Schrödinger operators on fibre bundles (Q522643)

Let \(\pi:M\rightarrow B\) be a fibre bundle of smooth manifolds with boundary, where the fibre \(F\) is compact and the base \(B\) is complete. Let \(g\) and \(g_B\) be Riemannian metrics on \(M\) and \(B\) respectively. Let \(TF:=\text{ker}(\pi_\star)\) be the vertical subbundle of \(TM\) such that \(\pi_\star\) induces an isometry \(TM/TF\rightarrow TB\). Write \(g=g_F+\pi^\star g_B\) where \(g_F\) vanishes on the horizontal subbundle \(NF:=TF^\perp\). For \(\epsilon\ll 1\), consider the family of metrics \[ g_\epsilon:=g_F+\epsilon^{-2}\pi^\star g_B \] and the associated Schrödinger operator (with \(D(H)\subset \mathcal{H}:=L^2(M,g)\)) \[ H:=-\Delta_{g_\epsilon}+V+\epsilon H_1 \] where \(V\) is a potential and \(H_1\) a second order differential operator. The main result in the paper under review is stated, under some technical conditions on \(H\) (condition 1) and on the geometry of \(M\) (condition 2), as follows (Theorem 2.1): For every \(\Lambda>0\) and \(N\in\mathbb{N}\) there exists an orthogonal projection \(P_\epsilon\in\mathcal{L}(\mathcal{H})\cap\mathcal{L}(D(H))\) that satisfies \[ \mid\mid[ H,P_\epsilon]\rho(H)\mid\mid_{\mathcal{L}(\mathcal{H})}=\mathcal{O}(\epsilon^{N+1}) \] for every Borel function \(\rho:\mathbb{R}\rightarrow[ 0;1]\) with support in \((-\infty,\Lambda]\). Furthermore \(P_\epsilon-P_0=\mathcal{O}(\epsilon)\) in \(\mathcal{L}(D(H))\) and there exists a unitary operator \(U_\epsilon\) on \(\mathcal{H}\) that maps \(P_0\mathcal{H}\) to the image of \(P_\epsilon\). As a byproduct, the authors prove that the image of \(P_\epsilon\) is almost-invariant under \(e^{-iHt}\), i.e there exists a subspace of \(L^2(M)\) that is invariant under \(H\) up to errors of order \(\epsilon^{N+1}\). Moreover, spectral and dynamical features of the operator \(H\) on this invariant subspace are discussed through a careful study of the asymptotics of \(H\).