arXiv subjects
Publications and source records attributed to S. A. Denisov.
In this work, we establish strong asymptotics of multiple orthogonal polynomials of the second type for Angelesco systems with measures that satisfy Szegő conditions. We consider multi-indices that converge to infinity in the non-marginal directions.
We characterize even measures $μ=wdx+μ_s$ on the real line with finite entropy integral $\int_{R} \frac{\log w(t)}{1+t^2}dt>-\infty$ in terms of $2\times 2$ Hamiltonian generated by $μ$ in the sense of inverse spectral theory. As a corollary, we obtain criterion for spectral measure of Krein string to have converging logarithmic integral.
The conjecture by Steklov was solved negatively by Rakhmanov in 1979. His original proof was based on the formula for orthogonal polynomial obtained by adding point masses to the measure of orthogonality. In this note, we show how this polynomial can be obtained by applying the method developed recently for proving the sharp lower bounds for the problem by Steklov.
We prove that 3-dimensional Schrodinger operator with slowly decaying potential has an absolutely continuous spectrum that fills the positive half-line. The asymptotics of Green's function is obtained as well.
We prove that the massless Dirac operator in $\mathbb{R^3}$ with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.
Discovered by M.G.Krein analogy between polinomials orthogonal on the unit circle and generalized eigenfunctions of certain differential systems is used to obtain some new results in the spectral analysis of Sturm-Liouville operators. Some properties of Dirac-type differential systems are studied.