A magnetic Schrodinger operator on a periodic graph

作者：Badanin A V^*; Korotyaev E L

来源：Sbornik: Mathematics , 2010, 201(10): 1403-1448.

DOI：10.1070/SM2010v201n10ABEH004116

• periodic graph
• magnetic Schrodinger operator
• spectral bands
• asymptotic behaviour of spectral bands

This paper looks at a magnetic Shrodinger operator on a graph of special form in 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 vertical bar G(n)vertical bar -> infinity as n -> infinity. 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 -> 0 is described.

• 出版日期2010