Given two selfadjoint operators H 0 and V = V + ?V ? , we study the motion of the spectrum of the operator H() = H 0 + V as increases. Let be a real number. We consider the quantity (; H(); H 0) deened as a generalization of Krein's spectral shift function of the pair H(); H 0. We study the as-ymptotic behavior of (; H(); H 0) as ! 1: Applications to… (More)
We study the properties of Schrödinger operators −∆ ± V. We prove that if their negative spectra are discrete, then their positive spectra do not have gaps.
We study the spectral properties of Jacobi matrices. By using " higher order " trace formulae we obtain a result relating the properties of the elements of Jacobi matrices and the corresponding spectral measures. Complicated expressions for traces of some operators can be magically simplified allowing us to apply induction arguments. Our theorems are… (More)
1. In this short paper we extend the method of Laptev-Naboko-Safronov . New estimates for the discrete spectrum obtained in this paper allow one to prove stronger results compared to . The main technical tool of the paper  is the so called trace inequality, which relates properties of negative eigenvalues to the properties of the a.c. spectrum.… (More)
For a large class of multi-dimensional Schrödinger operators it is shown that the absolutely continuous spectrum is essentially supported by [0, ∞). We require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non-negative bounded potential outside the cone. In particular, we do not require the existence of… (More)
We study the distribution of eigenvalues of the one-dimensional Schrödinger operator with a complex valued potential V. We prove that if |V | decays faster than the Coulomb potential, then the series of imaginary parts of square roots of eigenvalues is convergent. Let V : [0, ∞) → C be a complex valued potential. The object of our investigation is the… (More)
acting in the space L 2 (R d). We study the relation between the behavior of V at the infinity and the properties of the negative spectrum of H. According to the Cwikel-Lieb-Rozenblum estimate , , the number of negative eigenvalues of H satisfies the relation (1.1) N ≤ C V d/2 dx, d ≥ 3, where C is independent of V. Similar estimates hold in… (More)