Pólya's conjecture for Dirichlet eigenvalues of spherical shells and Aharonov--Bohm annuli

Abstract

We establish strict Dirichlet Pólya inequalities for spherical shells in every dimension $d\ge2$ and for every radius ratio. The inclusive eigenvalue counting function lies strictly below the leading Weyl term at every positive energy; consequently, each eigenvalue strictly exceeds its Weyl prediction, with multiplicities included. We also prove the same strict inequality for the Aharonov--Bohm operator on planar annuli at arbitrary flux. In dimension two, the nonmagnetic result gives a fully analytic proof of the planar annulus theorem of Filonov, Levitin, Polterovich and Sher (J. Lond. Math. Soc., 2026). The common ingredient is a lattice-point bound for the shell action, uniform in the starting point of a tail and in the lattice translation. Quadrature for the inverse action controls the change of convexity caused by the inner boundary. The tail estimate supplies the hypothesis of the known dimension-reduction argument for balls, while lattice translations represent the magnetic flux. Laptev's product argument yields corresponding strict bounds on products of shells and nonempty bounded open sets.