Spectrum of Multidimensional Schrödinger Operators with Sparse Potentials

The paper contains analysis of the spectrum of multidimensional Schrödinger operators on the lattice ℤ ^d and in ℜ d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429186875/b86f08e4-14d7-465f-ac8c-bce3ede29963/content/inq_chapter19_231_1.tif"/> with sparse potentials. The latter means that the potential is an infinite sum of bumps with distances between their supports going to infinity or with the density of the bumps decaying at infinity. Among other results, the coexistence of absolutely continuous and pure point spectrum is shown and an analog of Kato theorem is proven when potential decays at infinity only in average. It is also shown that in this case, there may exist eigenvalues embedded into continuous spectrum.