Schrödinger operators with periodic potentials and constant magnetic fields with integer flux (Q1382134)

The author investigates spectral properties of the two-dimensional Schrödinger operator \[ H(\lambda) =\Bigl(\textstyle {{1\over i}} \nabla+ a\Bigr)^2 +\lambda^2V \quad \text{in } L^2 (\mathbb{R}^2), \] where \(a(x_1,x_2) =b(x_2,-x_1)\), \(b\in \mathbb{R}\), \(\lambda\) is a positive parameter and \(V\) is a real-valued function which is periodic with respect to \(\Gamma =2 \pi \mathbb{Z} \oplus 2\mathbb{Z}\). Other conditions are imposed on \(V\) and it is assumed that \(b\in (1/4\pi) \mathbb{Z}\). The spectral properties depend critically on the properties of \(\Gamma\) and the magnetic field, which is defined by the 2-form \(B=- 2bdx_1 \wedge dx_2\); in particular the spetrum of \(H(\lambda)\) has a band structure. Two theorems are proved concerning the width \(L(\lambda)\) of the lowest band. In the first (Theorem A) it is proved that for any \(\eta>0\) there exists a constant \(C_\eta>0\) such that as \(\lambda \to\infty\) \[ L(\lambda)\leq C_\eta e^{-(s_0-2 \eta) \lambda}, \] where \(s_0\) is determined by the Agmon metric associated with \(V\). Under additional conditions, it is shown in Theorem B that as \(\lambda\to \infty\) \[ L (\lambda)= \bigl(b \lambda^{3/2} +O(\lambda^{1/2}) \bigr)e^{-s_0 \lambda} \] where \(b>0\) is a constant depending only on \(V\) and \(B\). The proofs are based on work of Helffer-Sjöstrand and Outassourt.