Michael Levitin

Pólya’s conjecture for the disk: a computer-assisted proof

The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. Pólya’s conjecture is known to be true for domains which tile the space, and, in addition, for some special domains in higher dimensions, but is still open in other cases. I’ll present a recent joint work with Iosif Polterovich and David Sher in which we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems, and, rather surprisingly, obtain some new results on the classical Bessel functions. Our proofs of Pólya’s conjecture are purely analytic for the values of the spectral parameter above some large (but explicitly given) number. Below that number we give a rigorous computer-assisted proof which converges in a finite number of steps and uses only integer arithmetic.