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SUMMARY:Bernoulli lecture IV - Eigenvalue distribution for non-self-adjoin
 t differential operators\, with and without analyticity.
DTSTART:20131210T171500
DTEND:20131210T181500
DTSTAMP:20260407T144150Z
UID:d396bc4c031785a015f215315a5d7f73e2425d3d7a7a5da79a5acb6a
CATEGORIES:Conferences - Seminars
DESCRIPTION:Johannes Sjostrand\nFor self-adjoint differential operators\, 
 a classical result of H.Weyl and its extensions since a century\, give a v
 ery general law for the asymptotic distribution of eigenvalues in the high
  energy and in the semi-classical limits. This law describes the density o
 f eigenvalues in terms of the direct image of the symplectic volume elemen
 t on (the real) phase space under a symbol map.\nOn the other hand\, for d
 ifferential operators in one dimension with analytic coefficients\, the co
 mplex WKB-method is a powerful tool to study the spectrum. In the non-self
 -adjoint case\, this often leads to a complex Bohr-Sommerfeld rule for the
  eigenvalues that invalidates the Weyl law.\nIn higher dimensions\, simila
 r results can be obtained when the coefficients are analytic and the compl
 ex WKB-method is then replaced by analytic microlocal analysis and deforma
 tions of the real phase space. In 2 dimensions\, one often encounters a 
 “hidden” complete integrability\, related to an old theorem of Cherry.
  Small non-self-adjoint perturbations of self-adjoint differential operato
 rs have their spectra in thin bands and quite detailed results are availab
 le in the integrable case.\nNon-self-adjoint operators do most of the time
  exhibit large resolvent norms and corresponding eigenvalue instability. I
 t is then natural to study the effect of small random perturbations. It ca
 me as a surprise that such perturbations cause the eigenvalues to distribu
 te according to the Weyl law. An heuristic explanation is that the random 
 perturbations destroy analyticity and forbid complex deformations of phase
  space\, leaving the Weyl law as the only natural possibility.\nWe will st
 ate some results and develop the underlying ideas of complex phase space d
 eformations and random perturbations\, following works of A. Melin\, M. Hi
 trik\, S. Vu Ngoc\, M. Hager\, W. Bordeaux Montrieux\, the speaker and oth
 er people.
LOCATION:BI A0 478
STATUS:CONFIRMED
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