Pólya’s conjecture in spectral geometry (1954) 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 (March 2022) 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 using some new relations for Bessel functions. Our proofs of Pólya’s conjecture for both Dirichlet and Neumann problems are purely analytic for the values of the spectral parameter above some large (but explicitly given) number. Below that number we give rigorous computer-assisted proofs which converge in a finite number of steps and use only integer arithmetic. I will also discuss some more recent improvements obtained by Filonov (August 2022).
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