On the tunneling effect for magnetic Schrödinger operators in antidot lattices

Article type: Research Article

Authors: Frank, Rupert L.

Affiliations: Royal Institute of Technology, Department of Mathematics, Lindstedtsvägen 25, S-100 44 Stockholm, Sweden E-mail: [email protected]

Abstract: We study the Schrödinger operator (hD−A)2 with periodic magnetic field B=curl A in an antidot lattice $\varOmega_{\infty}=\mathbb{R}^{2}\setminus\bigcup_{\alpha\in\varGamma}(U+\alpha)$. Neumann boundary conditions lead to spectrum below hinf B. Under suitable assumptions on a “one-well problem” we prove that this spectrum is localized inside an exponentially small interval in the semi-classical limit h→0. For this purpose we construct a basis of the corresponding spectral subspace with natural localization and symmetry properties.

Keywords: semi-classical analysis, tunneling effect, magnetic Schrödinger operator, periodic operator

Journal: Asymptotic Analysis, vol. 48, no. 1-2, pp. 91-120, 2006