Michael Levitin University of Reading

From asymptotics to bounds in spectral geometry and elsewhere


Suppose that we have an asymptotic expansion of some parameter-dependent quantity as parameter tends to infinity. It may happen that truncating such an expansion after a finite number of terms produces a bound for the quantity in question valid for all (not necessarily large) values of the parameter. A classical example of such a phenomenon is Pólya’s conjecture in spectral geometry which states that the leading Weyl term of the asymptotics of the eigenvalue counting function for the Dirichlet or Neumann Laplacian in a Euclidean domain gives a uniform upper or lower bound, respectively, for the counting function everywhere. I will discuss this and related problems and state some open questions. The talk is based on joint works with N. Filonov, I. Polterovich, and D. A. Sher.