On the asymptotic number of low-lying states in the two-dimensional confined Stark effect

We investigate the Stark operator restricted to a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with Dirichlet boundary conditions. In the semiclassical limit, a three-term asymptotic expansion for its individual eigenvalues has been established, with coefficients dependent on the curvature of $\mathrm{\Omega }$. We analyse the accumulation of eigenvalues beneath the leading-order terms in these expansions, establishing Weyl-type asymptotics. Furthermore, we derive weak asymptotics for the density of the spectral projector onto these low-lying states.