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SUMMARY:Periodic spectral problem for the massless Dirac operator - Vassil
 iev\, D (University College London)
DTSTART:20150325T150000Z
DTEND:20150325T160000Z
UID:TALK58569@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Co-author: Michael Levitin (University of Reading) \n\nPeriodi
 c spectral problems are normally formulated in terms of the Schrodinger op
 erator. The aim of the talk is to examine issues that arise if one formula
 tes a periodic spectral problem in terms of the Dirac operator. \n\nThe mo
 tivation for the particular model considered in the talk does not come fro
 m solid state physics. Instead\, we imagine a single massless neutrino liv
 ing in a compact 3-dimensional universe without boundary. There is no elec
 tromagnetic field in our model because a neutrino does not carry an electr
 ic charge and cannot interact (directly) with an electromagnetic field. Th
 e role of the electromagnetic covector potential is therefore taken over b
 y the metric. In other words\, we are interested in understanding how the 
 curvature of space affects the energy levels of the neutrino. \n\nMore spe
 cifically\, we consider the massless Dirac operator on a 3-torus equipped 
 with Euclidean metric and standard spin structure. It is known that the ei
 genvalues can be calculated explicitly: the spectrum is symmetric about ze
 ro and zero itself is a double eigenvalue. Our aim is to develop a perturb
 ation theory for the eigenvalue with smallest modulus with respect to pert
 urbations of the metric. Here the application of perturbation techniques i
 s hindered by the fact that eigenvalues of the massless Dirac operator hav
 e even multiplicity\, which is a consequence of this operator commuting wi
 th the antilinear operator of charge conjugation (a peculiar feature of di
 mension 3). We derive an asymptotic formula for the eigenvalue with smalle
 st modulus for arbitrary perturbations of the metric and present two parti
 cular families of Riemannian metrics for which the eigenvalue with smalles
 t modulus can be evaluated explicitly. We also establish a relation betwee
 n our asymptotic formu la and the eta invariant. \n\n[1] R.J.Downes\, M.Le
 vitin and D.Vassiliev\, Spectral asymmetry 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
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