Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators

We establish necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with a positive scalar potential. They are expressed in terms of Wiener capacity and local energy of the magnetic field. These conditions depend on two functional parameters. One of them is a decreasing function of one variable whose argument is the normalized local energy of the magnetic field. This function enters the negligibility condition of sets for the scalar potential. We almost precisely describe all admissible functional parameters. The corresponding conditions for different admissible functions are equivalent, which follows from the fact that any of them is equivalent to the discreteness of spectrum. Research partially supported by Oberwolfach Forschungsinstitut für Mathematik Research partially supported by NSF grant DMS-0107796