tailieunhanh - Đề tài " Discreteness of spectrum and positivity criteria for Schr¨odinger operators "

We provide a class of necessary and sufficient conditions for the discreteness of spectrum of Schr¨dinger operators with scalar potentials which o are semibounded below. The classical discreteness of spectrum criterion by A. M. Molchanov (1953) uses a notion of negligible set in a cube as a set whose Wiener capacity is less than a small constant times the capacity of the cube. We prove that this constant can be taken arbitrarily between 0 and 1. This solves a problem formulated by I. M. Gelfand in 1953. Moreover, we extend the notion of negligibility by allowing the constant to. | Annals of Mathematics Discreteness of spectrum and positivity criteria for Schr odinger operators By Vladimir Maz ya and Mikhail Shubin Annals of Mathematics 162 2005 919 942 Discreteness of spectrum and positivity criteria for Schrodinger operators By Vladimir MAz yA and Mikhail Shubin Abstract We provide a class of necessary and sufficient conditions for the discreteness of spectrum of Schrodinger operators with scalar potentials which are semibounded below. The classical discreteness of spectrum criterion by A. M. Molchanov 1953 uses a notion of negligible set in a cube as a set whose Wiener capacity is less than a small constant times the capacity of the cube. We prove that this constant can be taken arbitrarily between 0 and 1. This solves a problem formulated by I. M. Gelfand in 1953. Moreover we extend the notion of negligibility by allowing the constant to depend on the size of the cube. We give a complete description of all negligibility conditions of this kind. The a priori equivalence of our conditions involving different negligibility classes is a nontrivial property of the capacity. We also establish similar strict positivity criteria for the Schrodinger operators with nonnegative potentials. 1. Introduction In 1934 K. Friedrichs 3 proved that the spectrum of the Schrodinger operator A V in L2 Rn with a locally integrable potential V is discrete provided V x to as x TO see also 1 11 . On the other hand if we assume that V is semi-bounded below then the discreteness of spectrum easily implies that for every d 0 V x dx to as Qd TO JQd where Qd is an open cube with the edge length d and with the edges parallel to coordinate axes and Qd TO means that the cube Qd goes to infinity with fixed d . This was first noticed by A. M. Molchanov in 1953 see 10 who also The research of the first author was partially supported by the Department of Mathematics and the Robert G. Stone Fund at Northeastern University. The research of the second author was partially .