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Resonances for thin barriers on the circle

Galkowski, J; (2016) Resonances for thin barriers on the circle. Journal of Physics A: Mathematical and Theoretical , 49 (12) , Article 125205. 10.1088/1751-8113/49/12/125205. Green open access

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Abstract

We study high energy resonances for the operator-ΔV, ∂ω :=- Δ +ΔV, ∂ω ⊗ V when V has strong frequency dependence. The operator -ΔV, ∂ω is a Hamiltonian used to model both quantum corrals (Aligia and Lobos 2005 J. Phys.: Condens. Matter 17 S1095, Barr 2010 Nano Lett. 10 325360) and leaky quantum graphs (Exner P 2008 Analysis on Graphs and its Applications (Providence, RI: American Mathematical Society) pp 523564). Since highly frequency dependent delta potentials are out of reach of the more general techniques in (Galkowski J 2015 arXiv:1511.05894, Galkowski J and Smith H 2014 Int. Math. Res. Not.) we study the special case where ω = B(0, 1) CR2 and V =h-αV0 > 0 with α1. Here h-1 ∼ Re∇is the frequency. We give sharp bounds on the size of resonance free regions for α≤ 1 and the location of bands of resonances when 5/6≤α≤ 1. Finally, we give a lower bound on the number of resonances in logarithmic size strips: -MlogRe∇≤Im∇≤0.

Type: Article
Title: Resonances for thin barriers on the circle
Open access status: An open access version is available from UCL Discovery
DOI: 10.1088/1751-8113/49/12/125205
Publisher version: https://doi.org/10.1088/1751-8113/49/12/125205
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: resonances, quantum corral, delta potential, high energy, leaky quantum graph
UCL classification: UCL
UCL > Provost and Vice Provost Offices
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/10083911
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