A remark on Pólya’s conjecture at low frequencies

Abstract

We show that the Faber–Krahn inequality implies Polya’s conjecture for eigenvalues $$\lambda _{k}$$ of the Dirichlet Laplacian in $$\mathbb {R}^n$$ up to $$k = \lfloor b(n) \rfloor $$ , where b(n) is a function with exponential growth on the dimension. This function also appears in Pleijel’s bound for the number of nodal domains of the sequence of eigenfunctions of the ball and we improve on previous estimates by providing precise upper and lower bounds for b which coincide up to the first four terms in the expansion of $$\log (b(n))$$ for large n.