Spectrum of the Landau Hamiltonian with a periodic electric potential

L. I. Danilov

Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 1, pp. 47-65

Résumé

We define a class of periodic electric potentials for which the spectrum of the two-dimensional Schrödinger operator is absolutely continuous in the case of a homogeneous magnetic field $B$ with a rational flux $\eta= (2\pi)^{-1}Bv(K)$, where $v(K)$ is the area of an elementary cell $K$ in the lattice of potential periods. Using properties of functions in this class, we prove that in the space of periodic electric potentials in $L^2_{\mathrm{loc}}(\mathbb R^2)$ with a given period lattice and identified with $L^2(K)$, there exists a second-category set (in the sense of Baire) such that for any electric potential in this set and any homogeneous magnetic field with a rational flow $\eta$, the spectrum of the two-dimensional Schrödinger operator is absolutely continuous.

Détail

Keywords: two-dimensional Schrödinger operator, absolute spectrum continuity, periodic potential, homogeneous magnetic field.

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