arXiv subjects

Alexei Poltoratski

Publications and source records attributed to Alexei Poltoratski.

At least 19 recordsLinked to original sources

arXiv2026

Complex Analysis in completeness, spectral and scattering problems

These lectures are aimed at junior researchers seeking to enter the field of applications of complex analysis to harmonic analysis, Fourier analysis, spectral and scattering theory for differential operators and related fields. Starting with classical completeness problems posed by Bernstein, Beurling and Malliavin, Wiener, Kolmogorov and Krein, the course progresses to the solutions of the Gap and Type Problems, inverse spectral problems for Schr\"odinger operators and canonical Hamiltonian systems, and pointwise convergence of the non-linear Fourier transform. Each lecture is supplemented with exercises and references.

math.SP↗

arXiv2026

A maximal estimate for the non-linear Fourier transform

We discuss estimates for the maximal operator associated with the non-linear Fourier transform of an $L^2$-function on the half-line. For potentials with bounded dyadic $L^1$ masses we prove weak-type maximal and maximal fluctuation estimates on the sets where the spectral densities are bounded away from zero and three classical maximal functions of the spectral data are uniformly small; such sets exhaust almost all of the real line as the parameters of their definition relax.

math.CV↗

arXiv2026

Problems in spectral analysis of canonical Hamiltonian systems

This note focuses on recent results in spectral analysis of canonical systems of differential equations obtained via the approach developed in our previous papers \cite{MIF1, MP3, etudes, etudes2, PZ, Direct}. Many of our results are motivated by the pioneering research of Barry Simon and his co-authors; see, for instance, the papers cited in the main text. We discuss solutions to the inverse spectral problem (ISP) for canonical Hamiltonian systems and mixed spectral problems for Schr\"odinger operators. One of our goals is to show connections of ISP with classical tools of analysis, such as the Hilbert transform, orthogonal polynomials, the gap problem and solutions to the Riemann-Hilbert problem. We illustrate our results with examples and discuss further questions.

math.SP↗

arXiv2025

Etudes in the inverse spectral problem, II

We apply the approach developed in our previous papers to obtain examples of solutions to the inverse spectral problem (ISP) for the canonical Hamiltonian system. One of our goals is to illustrate connections of ISP with classical tools of analysis, such as the Hilbert transform and solutions to the Riemann-Hilbert problem. A key role in our study is played by the systems with homogeneous and quasi-homogeneous spectral measures. We show how some of such systems give rise to families of Bessel functions.

math.CV↗

arXiv2023

Etudes for the inverse spectral problem

In this note we study inverse spectral problems for canonical Hamiltonian systems, which encompass a broad class of second order differential equations on a half-line. Our goal is to extend the classical resultss developed in the work of Marchenko, Gelfand-Levitan, and Krein to broader classes of canonical systems and to illustrate the solution algorithms and formulas with a variety of examples. One of the main ingredients of our approach is the use of truncated Toeplitz operators, which complement the standard toolbox of the Krein-de Branges theory of canonical systems.

math.SP↗

arXiv2022

Periodic approximations in inverse spectral problems for canonical Hamiltonian systems

This note is devoted to inverse spectral problems for canonical Hamiltonian systems on the half-line. An approach to inverse spectral problems based on the use of truncated Toeplitz operators has been especially effective in the case when the spectral measure of the system is a locally finite periodic measure (see \cite{MP}). In this note we extend the periodic algorithm to the case of non-periodic measures by considering periodizations of a spectral measure and showing that the Hamiltonians corresponding to the periodizations converge to the Hamiltonian of the original measure.

math.SP↗

arXiv2021

Pointwise convergence of the non-linear Fourier transform

We prove pointwise convergence for the scattering data of a Dirac system of differential equations. Equivalently, we prove an analog of Carleson's theorem on almost everywhere convergence of Fourier series for a version of the non-linear Fourier transform. Our proofs are based on the study of resonances of Dirac systems.

math.CV↗

arXiv2018

Type alternative for Frostman measures

For a finite positive Borel measure $μ$ on $\mathbb R$ its exponential type, $T_μ$, is defined as the infimum of $a>0$ such that finite linear combinations of complex exponentials with frequencies between 0 and $a$ are dense in $L^2(μ)$. The definition can be easily extended from finite to broader classes of measures. In this paper we prove a new formula for $T_μ$ and use it to study growth and additivity properties of measures with finite positive type. As one of the applications, we show that Frostman measures on $\mathbb R$ may only have type zero or infinity.

math.CA↗

arXiv2018

Toeplitz Order

A new approach to problems of the Uncertainty Principle in Harmonic Analysis, based on the use of Toeplitz operators, has brought progress to some of the classical problems in the area. The goal of this paper is to develop and systematize the function theoretic component of the Toeplitz approach by introducing a partial order on the set of inner functions induced by the action of Toeplitz operators. We study connections of the new order with some of the classical problems and known results. We discuss remaining problems and possible directions for further research.

math.CV↗

arXiv2017

Two-Spectra Theorem with Uncertainty

The goal of this paper is to combine ideas from the theory of mixed spectral problems for differential operators with new results in the area of the Uncertainty Principle in Harmonic Analysis (UP). Using recent solutions of Gap and Type Problems of UP we prove a version of Borg's two-spectra theorem for Schrödinger operators, allowing uncertainty in the placement of the eigenvalues. We give a formula for the exact 'size of uncertainty', calculated from the lengths of the intervals where the eigenvalues may occur. Among other applications, we describe pairs of indeterminate operators in the three-interval case of the mixed spectral problem. At the end of the paper we discuss further questions and open problems.

math.SP↗

arXiv2015

De Branges functions of Schroedinger equations

We characterize the Hermite-Biehler (de Branges) functions $E$ which correspond to Shroedinger operators with $L^2$ potential on the finite interval. From this characterization one can easily deduce a recent theorem by Horvath. We also obtain a result about location of resonances.

math.CV↗

arXiv2014

Determinacy for measures

We study the general moment problem for measures on the real line, with polynomials replaced by more general spaces of entire functions. As a particular case, we describe measures that are uniquely determined by a restriction of their Fourier transform to a finite interval. We apply our results to prove an extension of a theorem by Eremenko and Novikov on the frequency of oscillations of measures with a spectral gap (high-pass signals) near infinity.

math.CA↗

arXiv2013

Cyclicity in rank-one perturbation problems

The property of cyclicity of a linear operator, or equivalently the property of simplicity of its spectrum, is an important spectral characteristic that appears in many problems of functional analysis and applications to mathematical physics. In this paper we study cyclicity in the context of rank-one perturbation problems for self-adjoint and unitary operators. We show that for a fixed non-zero vector the property of being a cyclic vector is not rare, in the sense that for any family of rank-one perturbations of self-adjoint or unitary operators acting on the space, that vector will be cyclic for every operator from the family, with a possible exception of a small set with respect to the parameter. We discuss applications of our results to Anderson-type Hamiltonians.

math-ph↗

arXiv2011

Bernstein's problem on weighted polynomial approximation

We formulate and discuss a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure $μ$ on the real line we give a criterion for density of polynomials in $L^p(μ)$.

math.CV↗

arXiv2011

A problem on completeness of exponentials

Let $μ$ be a finite positive measure on the real line. For $a>0$ denote by $\EE_a$ the family of exponential functions $$\EE_a=\{e^{ist}| \ s\in[0,a]\}.$$ The exponential type of $μ$ is the infimum of all numbers $a$ such that the finite linear combinations of the exponentials from $\EE_a$ are dense in $L^2(μ)$. If the set of such $a$ is empty, the exponential type of $μ$ is defined as infinity. The well-known type problem asks to find the exponential type of $μ$ in terms of $μ$. \ms\no In this note we present a solution to the type problem and discuss its relations with known results.

math.CA↗

arXiv2010

Approximation results for reflectionless Jacobi matrices

We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by periodic operators?

math.SP↗

arXiv2009

Polya sequences, Toeplitz kernels and gap theorems

A separated sequence $Λ$ on the real line is called a Polya sequence if any entire function of zero exponential type bounded on $Λ$ is constant. In this paper we solve the problem by Polya and Levinson that asks for a description of Polya sets. We also show that the Polya-Levinson problem is equivalent to a version of the so-called Beurling gap problem on Fourier transforms of measures. The solution is obtained via a recently developed approach based on the use of Toeplitz kernels and de Branges spaces of entire functions.

math.CV↗

arXiv2009

The Hilbert Transform of a Measure

Let $\fre$ be a homogeneous subset of $\bbR$ in the sense of Carleson. Let $μ$ be a finite positive measure on $\bbR$ and $H_μ(x)$ its Hilbert transform. We prove that if $\lim_{t\to\infty} t \abs{\fre\cap\{x\mid\abs{H_μ(x)}>t\}}=0$, then $μ_s(\fre)=0$, where $μ_\s$ is the singular part of $μ$.

math-ph↗