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  • 3 - The Laplacian on a Domain

  • from Part Two - The Laplace and Schrödinger Operators
  • Rupert L. Frank, Ludwig-Maximilians-Universität München, Ari Laptev, Imperial College of Science, Technology and Medicine, London, Timo Weidl, Universität Stuttgart
  • Book:
    Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities
    Published online:
    03 November 2022
    Print publication:
    17 November 2022, pp 167-243

  • Summary

    We discuss the definition of the Laplace operator on an open subset in Euclidean space as a self-adjoint operator with Dirichlet or Neumann boundary conditions and we derive its basic spectral properties. Among others, we include the spectral inequalities of Faber–Krahn, Hersch, and Friedlander. Then, using the technique of Dirichlet–Neumann bracketing, we derive Weyl's law for the asymptotic distribution of eigenvalues. We supplement this with a discussion of non-asymptotic bounds, including Pólya's conjecture and its proof for tiling domains and domains of product form. We present the sharp eigenvalue bounds of Berezin and Li–Yau. Finally, using separation of variables in spherical coordinates, we discuss the Laplacian on a ball.