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SUMMARY:Recent progress in the spectral theory of first order elliptic sys
 tems - Vassiliev\, D (University College London)
DTSTART:20120918T091000Z
DTEND:20120918T095000Z
UID:TALK39846@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:The talk deals with the distribution of eigenvalues of a linea
 r self-adjoint elliptic operator. The eigenvalue problem is considered in 
 the deterministic setting\, i.e. the coefficients of the operator are pres
 cribed smooth functions.  The objective is to derive a two-term asymptotic
  formula for the counting function (number of eigenvalues between zero and
  a positive lambda) as lambda tends to plus infinity.\n\nThere is an exten
 sive literature on the subject (see\, for example\, [1])\, mostly dealing 
 with scalar operators. It has always been taken for granted that all resul
 ts extend in a straightforward manner to systems. However\, the author has
  recently discovered [2\,3] that all previous publications on first order 
 systems give formulae for the second asymptotic coefficient that are eithe
 r incorrect or incomplete (i.e. an algorithm for the calculation of the se
 cond asymptotic coefficient rather than an explicit formula). The aim of t
 he talk is to explain the spectral theoretic difference between scalar ope
 rators and systems and to present the correct formula for the second asymp
 totic coefficient.\n\n[1] Yu.Safarov and D.Vassiliev\, The asymptotic dist
 ribution of eigenvalues of partial differential operators\, American Mathe
 matical Society\, 1997 (hardcover)\,\n1998 (softcover).\n\n[2] Preprint ar
 Xiv:1204.6567.\n\n[3] Preprint arXiv:1208.6015.\n
LOCATION:Seminar Room 1\, Newton Institute
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