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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 $$\Gamma_+$$.
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