The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrodinger operator is essentially supported by [0, infinity). Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables.

We discuss properties of eigenvalues of non-self-adjoint Schrodinger operators with complex-valued potential V. Among our results are estimates of the sum of powers of imaginary parts of eigenvalues by the L-p-norm of JV.

We obtain an asymptotic formula for the number of negative eigenvalues of a class of two-dimensional Schrodinger operators with small magnetic fields. This number increases as a coupling constant of the magnetic field tends to zero.

Let A be a periodic Schrödinger operator and let V ≥ 0 be a decaying potential. We study the number of the eigenvalues of the operator A(α)=A−α V inside a fixed interval (λ_{1},λ_{2}). We obtain an asymptotic formula for as α → ∞. In this paper we extend the results of Safronov (2001) for a more general class of perturbations.

We study the spectral properties of Jacobi matrices. By combining Killip's technique [12] with the technique of Killip and Simon [13] we obtain a result relating the properties of the elements of Jacobi matrices and the corresponding spectral measures. This theorem is a natural extension of a recent result of Laptev-Naboko-Safronov [17].