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Propagation through trapped sets and semiclassical resolvent estimates. (Propagation à travers des ensembles captés et estimations semiclassiques de la résolvante.) (English. French summary) Zbl 1271.58014
Ann. Inst. Fourier 62, No. 6, 2347-2377 (2012); addendum ibid. 62, No. 6, 2379-2384 (2012).
Denote \(R_h(\lambda)=(h^2\Delta+V-\lambda)^{-1}\) the resolvent of the Schrödinger operator on a complete Riemannian manifold \(X\) with real potential \(V\in C_0^\infty(X)\). The trapped set at energy \(E>0\) is the union of the relatively compact (maximally extended) bicharacteristic curves of \(|\xi|^2+V(x)=E\). The authors obtain semiclassical estimates for the truncated resolvent, \[ \|\chi R_h(E+i0)\chi\|={\mathcal O}(h^{-1}),\tag{\(\ast\)} \] when the cutoff \(\chi\) is microlocally supported away from the trapped set. An a priori polynomial bound is assumed, which means that an estimate \((\ast)\) with \({\mathcal O}(h^{-1})\) relaxed to \({\mathcal O}(h^{-k})\) holds. This condition does not hold in general, but it is satisfied if the trapping is sufficiently weak. Moreover, it is assumed that the resolvent is semiclassically outgoing, a condition which is satisfied if \(X\) has suitable ends at infinity, for instance, asymptotically conic or hyperbolic.
Resolvent estimates with non-microlocal cutoffs away from the projection of the trapped set were first obtained by N. Burq [Am. J. Math. 124, No. 4, 677–735 (2002; Zbl 1013.35019)]. Using a gluing technique of the authors [Int. Math. Res. Not. 2012, No. 23, 5409–5443 (2012; Zbl 1262.58019)], and a result of F. Cardoso and G. Vodev [Ann. Henri Poincaré 3, No. 4, 673–691 (2002; Zbl 1021.58016)], estimates \((\ast)\) are derived from a semiclassical propagation theorem which is the main result (Theorem 1.3) of the paper. Given \(\Gamma\Subset T^\ast X\) invariant under the bicharacteristic flow, there are defined the forward flowout \(\Gamma_+\) and the backward flowout \(\Gamma_-\) of \(\Gamma\). If \(u=R_h(\lambda)f\) is polynomially bounded, where \(f={\mathcal O}(1)\) and \(f\equiv 0\) semiclassically at \(\Gamma\), then, roughly speaking, Theorem 1.3 states that \({\mathcal O}(h^{-1})\)-bounds propagate from \(\Gamma_-\) to \(\Gamma_+\). The proof uses a positive commutator argument. The commutant equals the identity in a neighbourhood of \(\Gamma\) and decreases along \(\Gamma_+\).
See also the addendum [ibid. 62, No. 6, 2379–2384 (2012; Zbl 1271.58014)] to this paper.

MSC:
58J47 Propagation of singularities; initial value problems on manifolds
35L05 Wave equation
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