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- Ari Laptev, Oleg Safronov
- 2008

Throughout the paper, f± denotes either the positive or the negative part of f , which is, in its turn, either a function or a selfadjoint operator. The symbols <z and =z denote the real and the imaginary part of z. Finally, if a is a function on R, then a(i∇) is the operator whose integral kernel is (2π)−d ∫ eiξ(x−y)a(ξ)dξ. We consider the Schrödinger… (More)

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)

- Oleg Safronov
- 2009

where the constant C depends on t, γ and d (see also [6] for the case when V is real). The paper [5] deals with the natural question that appears in relation to the main result of [3]: what estimates are valid for the eigenvalues situated inside the conical sector {λ : |Iλ| < tRλ}, where the eigenvalues might be close to the positive half-line? Theorems of… (More)

1. In this short paper we extend the method of Laptev-Naboko-Safronov [15]. New estimates for the discrete spectrum obtained in this paper allow one to prove stronger results compared to [15]. The main technical tool of the paper [15] is the so called trace inequality, which relates properties of negative eigenvalues to the properties of the a.c. spectrum.… (More)

- Oleg Safronov
- 2000

In this paper we consider Schrödinger operators −∆+ V (x), V ∈ L∞(Rd) acting in the space L(R). If V = 0 then the operator has purely absolutely continuous spectrum on (0,+∞). We find conditions on V which guarantee that the absolutely continuous spectrum of both operators H+ = −∆ + V and H− = −∆− V is essentially supported by [0,∞). This means that the… (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)

- Oleg Safronov
- 2009

We study the distribution of eigenvalues of the Schrödinger operator with a complex valued potential V . We prove that if |V | decays faster than the Coulomb potential, then all eigenvalues are in a disc of a finite radius.

where C is independent of V . Similar estimates hold in dimensions d = 1 and d = 2. Therefore, if V decays at the infinity fast enough, then N is finite. The question arise if finiteness of N implies a qualified decay of V > 0 at the infinity? One can try to formulate theorems whose assumptions contain as less as possible information about V . But then it… (More)

- Oleg Safronov
- 2007

In dimension d ≥ 5, we consider the differential operator (1.1) H0 = −∆+ τζ(x)|x|(−∆θ), ε > 0, τ > 0, where ∆θ is the Laplace-Beltrami operator on the unit sphere S = {x ∈ R : |x| = 1} and ζ is the characteristic function of the complement to the unit ball {x ∈ R : |x| ≤ 1}. The standard argument with separation of variables allows one to define this… (More)