ON THE SEMI-CLASSICAL ANALYSIS OF SCHRÖDINGER OPERATORS WITH LINEAR ELECTRIC POTENTIALS ON A BOUNDED DOMAIN

Overview & Synopsis

The aim of this paper is to establish the asymptotic expansion of the eigenvalues of the Stark Hamiltonian, with 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, Krejcirik, 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.