Resonance-free regions for diffractive trapping by conormal potentials
| Autor: | Jared Wunsch, Oran Gannot |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
General Mathematics
Operator (physics) 010102 general mathematics Resonance Order (ring theory) 01 natural sciences 3. Good health Mathematics - Spectral Theory Hypersurface Compact space Mathematics - Analysis of PDEs Flow (mathematics) FOS: Mathematics Gravitational singularity 0101 mathematics Spectral Theory (math.SP) Energy (signal processing) Analysis of PDEs (math.AP) Mathematics Mathematical physics |
| Popis: | We consider the Schr\"odinger operator \[ P=h^2 \Delta_g + V \] on $\mathbb{R}^n$ equipped with a metric $g$ that is Euclidean outside a compact set. The real-valued potential $V$ is assumed to be compactly supported and smooth except at conormal singularities of order $-1-\alpha$ along a compact hypersurface $Y.$ For $\alpha>2$ (or even $\alpha>1$ if the classical flow is unique), we show that if $E_0$ is a non-trapping energy for the classical flow, then the operator $P$ has no resonances in a region \[ [E_0 - \delta, E_0 + \delta] - i[0,\nu_0 h \log(1/h)]. \] The constant $\nu_0$ is explicit in terms of $\alpha$ and dynamical quantities. We also show that the size of this resonance-free region is optimal for the class of piecewise-smooth potentials on the line. Comment: 20 pages; added Section 2.4 on applications to quantum evolution |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|