Propagation Through Trapped Sets and Semiclassical Resolvent Estimates

Abstract

AbstractLet \(P = -{h}^{2}\Delta + V (x)\), \(V \in {C}_{0}^{\infty }({\mathbb{R}}^{n})\).We are interested in semiclassical resolvent estimates of the form $$ \|\chi {(P - E - i0)}^{-1}{\chi \|}_{{L}^{2}({\mathbb{R}}^{n})\rightarrow {L}^{2}({\mathbb{R}}^{n})} \leq \frac{a(h)}{h} ,\qquad h \in (0,{h}_{0}],$$ (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 $${K}_{E} =\{ \alpha \in {T}^{{_\ast}}{\mathbb{R}}^{n}: \exists C > 0,\forall t > 0,\vert \exp (t{H}_{p})\alpha \vert \leq C\}?$$ Here \(p = \vert \xi {\vert }^{2} + V (x)\) and \({H}_{p} = 2\xi \cdot {\nabla }_{x} -\nabla V \cdot {\nabla }_{\xi }\).KeywordsInductive HypothesisGeneral OperatorGeneral VersusOpen CoverInductive StepThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.