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On the tunneling effect for magnetic Schrödinger operators in antidot lattices. (English) Zbl 1157.81309

Summary: We study the Schrödinger operator \((h{\mathbf{D}} - {\mathbf{A}})^2\) with periodic magnetic field \(B={\text{curl \textbf A}}\) in an antidot lattice \(\Omega_{\infty} = \mathbb{R}^{2} \setminus \bigcup_{\alpha \in \Gamma} (U+\alpha)\). Neumann boundary conditions lead to spectrum below \(h \inf B\). 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 \(h \to 0\). For this purpose we construct a basis of the corresponding spectral subspace with natural localization and symmetry properties.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47F05 General theory of partial differential operators
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