arXiv subjects
Publications and source records attributed to Sergey A. Denisov.
We study the connection between AAA (triple A) algorithm and multi-point Padé approximation when the approximated functions belong to either the Nevanlinna-Pick class or are Pólya frequency series.
We study strong asymptotics of orthogonal polynomials on an interval with varying weights, extending the framework of Totik's theorem on weighted orthogonal polynomials. Our results generalize this theorem in several directions. In particular, we allow the supports of the associated equilibrium measures to converge to a proper subinterval of the original interval of orthogonality or even collapse to a point. These extensions are motivated by applications to multiple orthogonality where such varying measures arise naturally.
We show that SU(1,1) NLFT can diverge pointwise for square-summable coefficients. As a consequence, we prove that the classical pointwise asymptotics of polynomials orthogonal on the unit circle can fail for measures in the Szegö class. We also discuss some special cases when the pointwise convergence holds.
We study the convergence of SU(1,1) and SU(2) nonlinear Fourier transform with sparse square-summable data. The asymptotics of the associated polynomials orthogonal on the unit circle is obtained as a corollary.
In the context of the Dirac equation with square-summable potential, we study the Jost solutions and prove that the maximal function associated with the argument of the transmission coefficient is unbounded. We also show that the strong version of the nonlinear Carleson conjecture fails for Dirac equations and Krein systems.
We study the wave operators for a Jacobi matrix whose spectral measure satisfies the Szegö condition. We prove existence and completeness of wave operators under a mild additional assumption on the Verblunsky coefficients of the associated measure on the unit circle.
This note compares two quantitative versions of the Nonlinear Carleson Conjecture (NCC). We provide motivations for our conjectures and show that they both imply the NCC. We discuss the connection to Carleson-Hunt maximal functions and give an SU(1,1) version of Calderon's theorem.
We develop the wave packet decomposition to study the Schrodinger evolution with rough potential. As an application, we obtain the improved bound on the wave propagation for the generic value of a parameter.
We apply inverse spectral theory to study Sobolev norms of solutions to the nonlinear Schrodinger equation. For initial datum $q_0\in L^2(\mathbb{R})$ and $s\in [-1,0]$, we prove that there exists a conserved quantity that is equivalent to $H^s(\mathbb{R})$-norm of the solution.
We study the spatial asymptotics of Green's function for the 1d Schrodinger operator with operator-valued decaying potential. The bounds on the entropy of the spectral measures are obtained. They are used to establish the presence of a.c. spectrum
We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality.
We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.
In three-dimensional case, we consider two classical operators: Schrodinger operator and an operator in the divergence form. For slowly-decaying oscillating potentials, we establish spatial asymptotics of the Green's function. The main term in this asymptotics involves vector-valued analytic function whose behavior is studied away from the spectrum. The absolute continuity of the spectrum is established as a corollary. For the operator in the divergence form, we consider the wave equation and establish existence of wave operators.
We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green's functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $\mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.
We consider the truncated equation for the centrally symmetric $V$-states in the 2d Euler dynamics of patches and prove the existence of solutions $y(x,λ),\, x\in [-1,1],\,λ\in (0,λ_0)$. These functions $y(x,λ)\in C^\infty(-1,1)$ and $\|y(x,λ)-|x|\|_{C[-1,1]}\to 0, λ\to 0$.
Some evolution equations with rough time-dependent potential are studied in the case of one-dimensional torus. We show that the solution has higher regularity for the generic values of the coupling constant. The asymptotics for large time and the localization on the Fourier side are studied.
For the 2d Euler dynamics of patches, we investigate the convergence to the singular stationary solutions in the presence of a regular strain. It is proved that the rate of merging can be made double exponential for all time.
For the two-dimensional Euler equation on the torus, we prove that the uniform norm of the vorticity gradient can grow as double exponential over arbitrarily long but finite time provided that at time zero it is already sufficiently large. Our result is equivalent to the statement that the Euler evolutions is linearly unbounded in Lipschitz norm for any time t>0.