Abstract. We prove that 3-dimensional Schrödinger operator with slowly decaying potential has an a.c. spectrum that fills R +. Asymptotics of Green’s functions is obtained as well. Consider the Schrödinger operator H = − ∆ + V, x ∈ R d We are interested in finding the support of an a.c. spectrum of H for the slowly decaying potential V. The following conjecture is due to B. Simon [21] Conjecture. If V (x) is such that R d

Abstract. We review recent advances in the spectral theory of Schrödinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994 International Congress on Mathematical Physics in Paris. The one-dimensional picture is now fairly complete, and provides many striking spectral examples. The multidimensional picture is still far from clear and may require deep original ideas for further progress. It might hold the keys for better understanding of a wide range of spectral and dynamical phenomena for Schrödinger operators in higher dimensions. 1.

Abstract. For a large class of Schrödinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are significantly easier to study. Then, we give some applications to the original Schrödinger operators including one-dimensional Schrödinger operators with L 2 – operator-valued potentials, multidimensional Schrödinger operators with slowly decaying potentials. 1.