Discreteness of spectrum for the magnetic Schr\"odinger operators. I
Abstract
We provide sufficient conditions for the discreteness of spectrum for magnetic Schr\"odinger operators. They generalize the classical result by K.Friedrichs (1934) and earlier results by J.Avron, I.Herbst and B.Simon (1978), A.Dufresnoy (1983) and A.Iwatsuka (1990) which were obtained in the absence of electric field. Our conditions are formulated as conditions of growth at infinity for some effective potentials. More precise sufficient conditions are formulated by use of the Wiener capacity and extend earlier work by A.M.Molchanov (1953) and V.Kondratiev and M.Shubin (1999) who considered the case of the operator without magnetic field. These conditions become necessary and sufficient in case when there is no magnetic field and the electric potential is semi-bounded below.
Keywords
Cite
@article{arxiv.math/0007111,
title = {Discreteness of spectrum for the magnetic Schr\"odinger operators. I},
author = {Vladimir Kondratiev and Mikhail Shubin},
journal= {arXiv preprint arXiv:math/0007111},
year = {2007}
}
Comments
43 pages