JournalsjstVol. 1, No. 1pp. 87–109

Geometrical Versions of improved Berezin–Li–Yau Inequalities

  • Leander Geisinger

    Universität Stuttgart, Germany
  • Ari Laptev

    Imperial College London, UK
  • Timo Weidl

    Universität Stuttgart, Germany
Geometrical Versions of improved Berezin–Li–Yau Inequalities cover
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Abstract

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in Rd\mathbb{R}^d, d2d \geq 2. In particular, we derive upper bounds on Riesz means of order σ3/2\sigma \geq 3/2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.

Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li–Yau inequality.

Cite this article

Leander Geisinger, Ari Laptev, Timo Weidl, Geometrical Versions of improved Berezin–Li–Yau Inequalities. J. Spectr. Theory 1 (2011), no. 1, pp. 87–109

DOI 10.4171/JST/4