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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Periodic spectral problem for the massless Dirac o
perator - Vassiliev\, D (University College London
)
DTSTART;TZID=Europe/London:20150325T150000
DTEND;TZID=Europe/London:20150325T160000
UID:TALK58569AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/58569
DESCRIPTION:Co-author: Michael Levitin (University of Reading)
\n\nPeriodic spectral problems are normally formu
lated in terms of the Schrodinger operator. The ai
m of the talk is to examine issues that arise if o
ne formulates a periodic spectral problem in terms
of the Dirac operator. \n\nThe motivation for the
particular model considered in the talk does not
come from solid state physics. Instead\, we imagin
e a single massless neutrino living in a compact 3
-dimensional universe without boundary. There is n
o electromagnetic field in our model because a neu
trino does not carry an electric charge and cannot
interact (directly) with an electromagnetic field
. The role of the electromagnetic covector potenti
al is therefore taken over by the metric. In other
words\, we are interested in understanding how th
e curvature of space affects the energy levels of
the neutrino. \n\nMore specifically\, we consider
the massless Dirac operator on a 3-torus equipped
with Euclidean metric and standard spin structure.
It is known that the eigenvalues can be calculate
d explicitly: the spectrum is symmetric about zero
and zero itself is a double eigenvalue. Our aim i
s to develop a perturbation theory for the eigenva
lue with smallest modulus with respect to perturba
tions of the metric. Here the application of pertu
rbation techniques is hindered by the fact that ei
genvalues of the massless Dirac operator have even
multiplicity\, which is a consequence of this ope
rator commuting with the antilinear operator of ch
arge conjugation (a peculiar feature of dimension
3). We derive an asymptotic formula for the eigenv
alue with smallest modulus for arbitrary perturbat
ions of the metric and present two particular fami
lies of Riemannian metrics for which the eigenvalu
e with smallest modulus can be evaluated explicitl
y. We also establish a relation between our asympt
otic formu la and the eta invariant. \n\n[1] R.J.D
ownes\, M.Levitin and D.Vassiliev\, Spectral asymm
etry of the massless Dirac operator on a 3-torus\,
Journal of Mathematical Physics\, 2013\, vol. 54\
, article 111503.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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