Kiril Datchev

Local Smoothing for Scattering Manifolds with Hyperbolic Trapped Sets

We prove a resolvent estimate for the Laplace-Beltrami operator on a scattering manifold with a hyperbolic trapped set, and as a corollary deduce local smoothing. We use a result of Nonnenmacher-Zworski to provide an estimate near the trapped region, a result of Burq and Cardoso-Vodev to provide an estimate near infinity, and the microlocal calculus on scattering manifolds to combine the two.

Propagation Through Trapped Sets and Semiclassical Resolvent Estimates

Let \(P = -{h}^{2}\Delta + V (x)\), \(V \in {C}_{0}^{\infty }({\mathbb{R}}^{n})\).We are interested in semiclassical resolvent estimates of the form

Resonance free regions for nontrapping manifolds with cusps

\|\chi {(P - E - i0)}^{-1}{\chi \|}_{{L}^{2}({\mathbb{R}}^{n})\rightarrow {L}^{2}({\mathbb{R}}^{n})} \leq \frac{a(h)}{h} ,\qquad h \in (0,{h}_{0}],

Resonant Uniqueness of Radial Semiclassical Schrödinger Operators

(1) for E > 0, \(\chi \in {C}^{\infty }({\mathbb{R}}^{n})\) with \(\vert \chi (x)\vert \leq \langle {x\rangle }^{-s}\), s > 1 ∕ 2. We ask: how is the function a(h) for which (1) holds affected by the relationship between the support of \(\chi \) and the trapped set at energy E, defined by

Gluing Semiclassical Resolvent Estimates via Propagation of Singularities

{K}_{E} =\{ \alpha \in {T}^{{_\ast}}{\mathbb{R}}^{n}: \exists C > 0,\forall t > 0,\vert \exp (t{H}_{p})\alpha \vert \leq C\}?

Quantitative Limiting Absorption Principle in the Semiclassical Limit

Here \(p = \vert \xi {\vert }^{2} + V (x)\) and \({H}_{p} = 2\xi \cdot {\nabla }_{x} -\nabla V \cdot {\nabla }_{\xi }\).