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