Semiclassical spectral bounds with remainder terms

Timo Weidl
Date and time
Conference - DimaScat : Scattering Theory and Spectral Asymptotics of Differential Operators - in Honour of Dimitri Yafaev

The Berezin and the Li-Yau inequalities state that the first term in Weyl's asymptotic formula serves also as uniform spectral bound on partial eigenvalue sums of the Dirichlet Laplacian. We shall report on various attempts to improve these bounds by taking terms of lower order into account. Moreover, we shall sketch some recent results on the magnetic Laplacian and on the Heisenberg Laplacian, respectively.

  1. H. Kovarik, T. Weidl: ``Improved Berezin-Li-Yau inequalities with magnetic fields''. Proceedings of the Royal Society of Edinburgh, 145A, 145-160, 2015
  2. H. Kovarik, B. Ruszkowski, T. Weidl: ``Melas-type bounds for the Heisenberg Laplacian on bounded domains'', to appear in Journal of Spectral Theory
  3. D. Barseghyan, P. Exner, H. Kovarik, T. Weidl: ``Semiclassical bounds in magnetic bottles'', to appear in Reviews in Mathematical Physics, 28 (1), 2016
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