P\’olya’s conjecture in spectral geometry and related questions.

Abstract: The 1954  conjecture of George P\’olya states that all the
eigenvalues of either the Dirichlet or the Neumann Laplacian on a
bounded Euclidean domain can be estimated, from below and above
respectively, by the leading term of their Weyl’s asymptotics: a
simple expression involving only the consecutive number of the
eigenvalue and the volume of the domain. Until recently it has been
known to be true in full generality only for domains which tile the
space. In a recent paper with N. Filonov, I. Polterovich, and D. Sher,
we proved P\’olya’s conjecture for Euclidean balls in any dimension
(the Dirichlet case) and in dimension two (Neumann case) as well as
for circular sectors of an arbitrary aperture; this is now in the
process of being extended to other problems, for example the magnetic
Aharonov—Bohm operator. The proofs rely on two important ingredients
from different fields which are of independent interest. The first one
comes from ODEs and is the new sharp uniform and relatively elementary
enclosures for the zeros of Bessel functions and of their derivatives
which surprisingly improve most of the earlier known bounds. The
second one is number-theoretical and concerns the bounds on the number
of shifted lattice points under the graph of a decreasing convex
function. Additionally, in the Neumann case we employ a rigorous
computer-assisted proof to close a small gap left by the theoretical
argument.