The high energy asymptotic distribution of the eigenvalues of the scattering matrix

Abstract

We determine the high energy asymptotic density of the eigenvalues of the scat- tering matrix associated with the operators H0 = −∆ and H = (i∇ + A)2 + V (x), where V : Rd → R is a smooth short-range real-valued electric potential and
A = (A1, . . . , Ad) : Rd → Rd is a smooth short-range magnetic vector-potential. Two cases are considered. The first case is where the magnetic vector-potential is non-zero. The spectral density of the associated scattering matrix in this case is expressed as an integral solely in terms of the magnetic vector-potential A. The second case considered is where the magnetic vector-potential is identically zero. Again the spectral density of the scattering matrix is expressed as an integral, this time in terms of the poten- tial V . These results share similar
characteristics to results pertaining to semiclassical asymptotics for pseudodifferential
operators.

Date of Award	Jun 2013
Original language	English
Awarding Institution	King's College London
Supervisor	Alexander Pushnitski (Supervisor), Eugene Shargorodsky (Supervisor) & Yuri Safarov (Supervisor)