Аннотация:
In 1911–1912 Hermann Weyl published two papers (more followed) describing the distribution of the eigenvalues of the Dirichlet Laplacian in a bounded domain. These were among the first publications by Weyl and a new exciting field of mathematics was created. I will discuss:
• Weyl law with sharper remainder estimates (in particular, Weyl conjecture)
• Generalized Weyl law
• When the generalized Weyl law works and when it does not and how it should be modified
• What should be used instead of the eigenvalue counting function when the spectrum is not necessarily discrete?
• Weyl law and Thomas-Fermi theory
• Some open problems.
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