Galkowski, J;
(2016)
The L² behavior of eigenfunctions near the glancing set.
Communications in Partial Differential Equations
, 41
(10)
pp. 1619-1648.
10.1080/03605302.2016.1227339.
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Abstract
Let M be a compact manifold with or without boundary and H⊂M be a smooth, interior hypersurface. We study the restriction of Laplace eigenfunctions solving (–h^{2} Δg – 1)u to H. In particular, we study the degeneration of u|H as one microlocally approaches the glancing set by finding the optimal power s0, so that (1 + h^{2} Δ_{H})^{S0}_{+}u|H remains uniformly bounded in L²(H) as h→0. Moreover, we show that this bound is saturated at every h-dependent scale near glancing using examples on the disk and sphere. We give an application of our estimates to quantum ergodic restriction theorems.
| Type: | Article |
|---|---|
| Title: | The L² behavior of eigenfunctions near the glancing set |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1080/03605302.2016.1227339 |
| Publisher version: | https://doi.org/10.1080/03605302.2016.1227339 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Eigenfunction estimates, pseudodifferential operators, restriction to hypersurfaces, semiclassical analysis, spectral weight |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10083906 |
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