Galkowski, J;
Shapiro, J;
(2021)
Semiclassical Resolvent Bounds for Long-Range Lipschitz Potentials.
International Mathematics Research Notices
, 2022
(18)
pp. 14134-14150.
10.1093/imrn/rnab134.
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Abstract
We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator −h^{2} Δ + V(x) − E in dimension n ≠ 2, where h,E > 0. The potential is real valued and V and ∂_{r} V exhibit long-range decay at infinity and may grow like a sufficiently small negative power of r as r → 0. The resolvent norm grows exponentially in h^{−1}, but near infinity it grows linearly. When V is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius CE^{−1/2} for some C > 0. This E-dependence is sharp and answers a question of Datchev and Jin.
| Type: | Article |
|---|---|
| Title: | Semiclassical Resolvent Bounds for Long-Range Lipschitz Potentials |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1093/imrn/rnab134 |
| Publisher version: | https://doi.org/10.1093/imrn/rnab134 |
| 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. |
| 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-pp.ucl.ac.uk/id/eprint/10128039 |
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