Yutian Li

P\'olya's Conjecture for the Neumann Eigenvalues on Euclidean Balls

We prove P\'olya's conjectured lower bound for the Neumann eigenvalue counting function of Euclidean balls. If $B_R^d\subset\mathbb R^d$ is the ball of radius $R$, then, for every $d\ge2$, $R>0$, and $E\ge0$, $$ N_{B_R^d}^{<}(E) \ge \frac{\omega_d}{(2\pi)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^d\Gamma(\frac d2+1)^2},...