Discreteness of spectrum for the magnetic Schrodinger operators

We consider a magnetic Schrodinger operator H in R-n or on a Riemannian manifold M of bounded geometry. Sufficient conditions for the spectrum of H to be discrete are given in terms of behavior at infinity for some effective potentials V-eff which are expressed through electric and magnetic fields. These conditions can be formulated in the form V-eff(x) --> infinity as x --> infinity. They generalize the classical result by K. Friedrichs (1934), and include earlier results of J. Avron, I. Herbst and B. Simon (1978), A. Dufresnoy (1983) and A. Iwatsuka (1990) which were obtained in the absence of an electric field, More precise sufficient conditions can be formulated in terms 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 a magnetic field. These conditions become necessary and sufficient in case there is no magnetic field and the electric potential is semi-bounded below.