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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:628250 |
| Date | January 2013 |
| Creators | Bulger, Daniel |
| Publisher | King's College London (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://kclpure.kcl.ac.uk/portal/en/theses/the-high-energy-asymptotic-distribution-of-the-eigenvalues-of-the-scattering-matrix(541fc908-ff77-4f0f-b3ba-af1fe53e19dd).html |
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