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Resonance-free regions for diffractive trapping by conormal potentials
- Publication Year :
- 2018
-
Abstract
- 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.<br />Comment: 20 pages; added Section 2.4 on applications to quantum evolution
- Subjects :
- Mathematics - Analysis of PDEs
Mathematics - Spectral Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1809.03012
- Document Type :
- Working Paper