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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Spectral theory of first order systems: an interfa
ce between analysis and geometry - Vassiliev\, D (
University College London)
DTSTART;TZID=Europe/London:20120803T114500
DTEND;TZID=Europe/London:20120803T123000
UID:TALK39166AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/39166
DESCRIPTION:We consider an elliptic self-adjoint first order p
seudodifferential operator acting on columns of co
mplex-valued half-densities over a connected compa
ct manifold without boundary. The eigenvalues of t
he principal symbol are assumed to be simple but n
o assumptions are made on their sign\, so the oper
ator is not necessarily semi-bounded. We study the
following objects: \n\na) the propagator (time-de
pendent operator which solves the Cauchy problem f
or the dynamic equation)\, \n\nb) the spectral fun
ction (sum of squares of Euclidean norms of eigenf
unctions evaluated at a given point of the manifol
d\, with summation carried out over all eigenvalue
s between zero and a positive lambda) and \n\nc) t
he counting function (number of eigenvalues betwee
n zero and a positive lambda). \n\nWe derive expli
cit two-term asymptotic formulae for all three. Fo
r the propagator "asymptotic" is understood as asy
mptotic in terms of smoothness\, whereas for the s
pectral and counting functions "asymptotic" is und
erstood as asymptotic with respect to the paramete
r lambda tending to plus infinity. In performing t
his analysis we establish that all previous public
ations on the subject are either incorrect or inco
mplete\, the underlying issue being that there is
simply too much differential geometry involved in
the application of microlocal techniques to system
s. \n\nWe then focus our attention on the special
case of the massless Dirac operator in dimension 3
and provide simple spectral theoretic characteris
ations of this operator and corresponding action (
variational functional). \n\n[1] O.Chervova\, R.J.
Downes and D.Vassiliev. The spectral function of a
first order system. Preprint arXiv:1204.6567.\n\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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