A~magnetic Schr\"odinger operator on a periodic graph

A. V. Badanin ; E. L. Korotyaev

Sbornik. Mathematics, Tome 201 (2010) no. 10, pp. 1403-1448

Résumé

This paper looks at a magnetic Shrödinger operator on a graph of special form in $\mathbb R^3$. It is called an armchair graph because graphs of this form with operators on them are used as a possible model for the so-called armchair nanotube in the homogeneous magnetic field which has amplitude $b$ and is parallel to the axis of the nanotube. The spectrum of the operator in question consists of an absolutely continuous part (spectral bands, separated by gaps) and finitely many eigenvalues of infinite multiplicity. The asymptotic behaviour of gaps for fixed $b$ and high energies is described; it is proved that for all values of $b$, apart from a discrete set containing $b=0$, there exists an infinite system of nondegenerate gaps $G_n$ with length $|G_n|\to\infty$ as $n\to\infty$. The dependence of the spectrum on the magnetic field is investigated and the existence of gaps independent of $b$ is proved for certain special potentials. The asymptotic behaviour of gaps as $b\to0$ is described. Bibliography: 32 titles.

Détail

Keywords: periodic graph, magnetic Schrödinger operator, spectral bands, asymptotic behaviour of spectral bands.