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.