Estimates for the number of eigenvalues of two dimensional Schr odinger operators lying below the essential spectrum

Abstract

The celebrated Cwikel-Lieb-Rozenblum inequality gives an upper estimate for the number of negative eigenvalues of Schr odinger operators in dimension three and higher. The situation is much more difficult in the two dimensional case. There has been significant progress in obtaining upper estimates for the number of negative eigenvalues of two dimensional Schrodinger operators on the whole plane. In this thesis, we present upper estimates of themCwikel-Lieb-Rozenblum type for the number of eigenvalues (counted with mmultiplicities) of two dimensional Schrodinger operators lying below the essential Spectrum in terms of the norms of the potential. The problem is considered on the whole plane with di_erent supports of the potential (in particular, sets of dimension _ 2 (0; 2] and on a strip with various boundary conditions. In both cases, the estimates involve weighted L1 norms and Orlicz norms of the potential.