Gluing semiclassical resolvent estimates via propagation of singularities

Analytic Continuation and Semiclassical Resolvent Estimates on Asymptotically Hyperbolic Spaces

“…This amounts to a change in the C ∞ structure of X; we denote the new manifold by X 1 2 . The second order elliptic operator ∆ X in ( 7) is self-adjoint, and non-negative, with respect to the measure (8) Ω = α −2 r n dr dω, with α given by (5). So, by the spectral theorem, the resolvent…”

“…This result is not known in higher dimensions (though the methods of Bony and Häfner would work even then), and to prove our main theorem we use the results of Datchev and the third author [5] and Wunsch and Zworski [24] to handle the general case. The advantage of the method of [5] is that one does not need to obtain a bound for the exact resolvent in the interior and we may work with the approximate model of [24] instead. We decompose the manifold X in two parts X = X 0 ∪ X 1 , where X 0 = [r bH , r bH + 4δ) × S n ∪ (r sI − 4δ, r sI ] × S n and X 1 = (r bH + δ, r sI − δ) × S n .…”

Extending Cutoff Resolvent Estimates via Propagation of Singularities

“…The same proof holds with only minor modifications when V is replaced by a metric perturbation or an obstacle or when χ is noncompactly supported but suitably decaying. As in [DaVa11], it can similarly treat suitably decaying perturbations or asymptotically hyperbolic manifolds in the sense of [Vas10,Vas11]. In particular, in the case a(h) = C log(1/h)/h above, a rescaling of parameters in the Theorem implies the conclusion of [Chr08, Corollary 2.3] in a more general trapping situation, namely we replace the assumption that there is one trapped orbit with (1.1).…”

“…In the present paper we use the gluing method via propagation of singularities developed in collaboration with Vasy in [DaVa11]; this is perhaps the simplest application of that method. The same proof holds with only minor modifications when V is replaced by a metric perturbation or an obstacle or when χ is noncompactly supported but suitably decaying.…”

“…I am very grateful to Vesselin Petkov and Nicolas Burq for suggesting this problem as an application of the gluing method of [DaVa11], and for helpful discussions and comments. Thanks also to the anonymous referee for several suggestions and corrections.…”