BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Spectral theory of first order systems: an interface between analy
 sis and geometry - Vassiliev\, D (University College London)
DTSTART:20120803T104500Z
DTEND:20120803T113000Z
UID:TALK39166@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We consider an elliptic self-adjoint first order pseudodiffere
 ntial operator acting on columns of complex-valued half-densities over a c
 onnected compact manifold without boundary. The eigenvalues of the princip
 al symbol are assumed to be simple but no assumptions are made on their si
 gn\, so the operator is not necessarily semi-bounded. We study the followi
 ng objects: \n\na) the propagator (time-dependent operator which solves th
 e Cauchy problem for the dynamic equation)\, \n\nb) the spectral function 
 (sum of squares of Euclidean norms of eigenfunctions evaluated at a given 
 point of the manifold\, with summation carried out over all eigenvalues be
 tween zero and a positive lambda) and \n\nc) the counting function (number
  of eigenvalues between zero and a positive lambda). \n\nWe derive explici
 t two-term asymptotic formulae for all three. For the propagator "asymptot
 ic" is understood as asymptotic in terms of smoothness\, whereas for the s
 pectral and counting functions "asymptotic" is understood as asymptotic wi
 th respect to the parameter lambda tending to plus infinity. In performing
  this analysis we establish that all previous publications on the subject 
 are either incorrect or incomplete\, the underlying issue being that there
  is simply too much differential geometry involved in the application of m
 icrolocal techniques to systems. \n\nWe then focus our attention on the sp
 ecial case of the massless Dirac operator in dimension 3 and provide simpl
 e spectral theoretic characterisations of this operator and corresponding 
 action (variational functional). \n\n[1] O.Chervova\, R.J.Downes and D.Vas
 siliev. The spectral function of a first order system. Preprint arXiv:1204
 .6567.\n\n
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR