On the semi-classical analysis of Schrödinger operators with linear electric potentials on a bounded domain

The aim of this paper is to establish the asymptotic expansion of the eigenvalues of the Stark Hamiltonian, with a strong uniform electric field and Dirichlet boundary conditions on a smooth bounded domain of R N , N ⩾ 2. This work aims at generalizing the recent results of Cornean, Krejčiřik, Pedersen, Raymond, and Stockmeyer in dimension 2. More precisely, in dimension N, in the strong electric field limit, we derive, under certain local convexity conditions, a full asymptotic expansion of the low-lying eigenvalues. To establish our main result, we perform the construction of quasi-modes. The “optimality” of our constructions is then established thanks to a reduction to model operators and localization estimates.