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

Rayan Fahs (1, 2)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 Faculté des Sciences, Université d'Angers

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.

Mots clés: Stark effects; strong electric field; asymptotics of eigenvalues; Dirichlet boundary conditions; Airy function; harmonic oscillator; Weingarten map

Domaines: Mathématiques [math]; Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Rayan Fahs. ON THE SEMI-CLASSICAL ANALYSIS OF SCHRÖDINGER OPERATORS WITH LINEAR ELECTRIC POTENTIALS ON A BOUNDED DOMAIN. Asymptotic Analysis, 2023, 135 (1-2), pp.81-113. ⟨10.3233/ASY-231848⟩. ⟨hal-03875344v2⟩