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

TL;DR

This paper studies the spectral properties of the two-dimensional Stark operator with Dirichlet boundary conditions, deriving asymptotic formulas for low-lying eigenvalues and their density in the semiclassical limit.

Contribution

It provides new Weyl-type asymptotics for eigenvalue accumulation and weak spectral projector asymptotics for the Stark operator in bounded domains.

Findings

Established Weyl-type asymptotics for eigenvalue accumulation.

Derived weak asymptotics for the spectral projector density.

Connected eigenvalue asymptotics to boundary curvature effects.

Abstract

We investigate the Stark operator restricted to a bounded domain $\Omega\subset\mathbb{R}^2$ 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 $\Omega$. 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.