5 problems

Let $Omega subset R^n$ be a sufficiently regular domain, and let $ND( Omega,t)$ denote its Dirichlet Laplace eigenvalue counting function. Write…

Baudoin–Tardif conjecture. The optimal convergence rate in this estimate converges to $\eta_1$ as $\gamma\to\infty$. The rate supplied by the existing exponential-convergence resul…

Let $\mathcal E_h=\{(x,y):(hx)^2+y^2<1\}$ be an ellipse with $h\in(0,1]$, and consider its Dirichlet or Neumann Laplace operator. Hezari–Zelditch's conjecture. The eigenvalues of a…

Let $P_{\gamma}=-\gamma X+\frac{1}{2}\gamma^2\Delta_{\mathbb{S}}$ be the generator of the rescaled kinetic Brownian motion on the unit sphere bundle $S\mathbb{M}$ over a compact Ri…

Let $({\mathbb M},g)$ be a compact Riemannian manifold, let $S{\mathbb M}$ be its unit sphere bundle, and let $P_{\gamma}=\gamma\widetilde P_{\gamma}$ be the generator of the resca…