eprintid: 1569444 rev_number: 17 eprint_status: archive userid: 608 dir: disk0/01/56/94/44 datestamp: 2017-10-30 10:06:21 lastmod: 2020-08-02 00:03:26 status_changed: 2017-10-30 10:06:21 type: article metadata_visibility: show creators_name: Datchev, K creators_name: Gell-Redman, J creators_name: Hassell, A creators_name: Humphries, P title: Approximation and Equidistribution of Phase Shifts: Spherical Symmetry ispublished: pub divisions: UCL divisions: A01 divisions: B04 divisions: C06 note: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. abstract: Consider a semiclassical Hamiltonian HV,h:=h2Δ+V−E, HV,h:=h2Δ+V−E, where h > 0 is a semiclassical parameter, Δ is the positive Laplacian on Rd,VRd,V is a smooth, compactly supported central potential function and E > 0 is an energy level. In this setting the scattering matrix Sh(E) is a unitary operator on L2(Sd−1)L2(Sd−1) , hence with spectrum lying on the unit circle; moreover, the spectrum is discrete except at 1. We show under certain additional assumptions on the potential that the eigenvalues of Sh(E) can be divided into two classes: a finite number ∼cd(RE−−√/h)d−1∼cd(RE/h)d−1 , as h→0h→0 , where B(0, R) is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder that are all very close to 1. Semiclassically, these are related to the rays that meet the support of, and hence are scattered by, the potential, and those that do not meet the support of the potential, respectively. A similar property is shown for the obstacle problem in the case that the obstacle is the ball of radius R. date: 2014-02 date_type: published official_url: http://dx.doi.org/10.1007/s00220-013-1841-8 oa_status: green full_text_type: other language: eng primo: open primo_central: open_green verified: verified_manual elements_id: 1410913 doi: 10.1007/s00220-013-1841-8 lyricists_name: Humphries, Peter lyricists_id: PHUMP20 actors_name: Humphries, Peter actors_id: PHUMP20 actors_role: owner full_text_status: public publication: Communications in Mathematical Physics volume: 326 number: 1 pagerange: 209-236 issn: 1432-0916 citation: Datchev, K; Gell-Redman, J; Hassell, A; Humphries, P; (2014) Approximation and Equidistribution of Phase Shifts: Spherical Symmetry. Communications in Mathematical Physics , 326 (1) pp. 209-236. 10.1007/s00220-013-1841-8 <https://doi.org/10.1007/s00220-013-1841-8>. Green open access document_url: https://discovery.ucl.ac.uk/id/eprint/1569444/1/Datchev_approximation_and_equidistribution_of_phase_shifts.pdf