56 citations · 549 across the 34 of their papers we have counts for
35 papers
Infinite-peakon solutions of the Camassa-Holm equation
Xiang-Ke Chang, Jonathan Eckhardt, Aleksey Kostenko
We study a class of (conservative) low regularity solutions to the Camassa-Holm equation on the line by exploiting the classical moment problem (in the framework of generalized ind…
Long-time asymptotics for the Korteweg-de Vries equation with integrable reflectionless initial data
Jonathan Eckhardt
We show that solutions of the Korteweg-de Vries equation with reflectionless integrable initial data decompose into a (in general infinite) linear superposition of solitons after l…
Trace formulas and inverse spectral theory for generalized indefinite strings
Jonathan Eckhardt, Aleksey Kostenko
Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and 2x2 canonica…
The conservative Camassa-Holm flow with step-like irregular initial data
Jonathan Eckhardt, Aleksey Kostenko
We extend the inverse spectral transform for the conservative Camassa-Holm flow on the line to a class of initial data that requires strong decay at one endpoint but only mild boun…
Generalized indefinite strings with purely discrete spectrum
Jonathan Eckhardt, Aleksey Kostenko
We establish criteria for the spectrum of a generalized indefinite string to be purely discrete and to satisfy Schatten-von Neumann properties. The results can be applied to the is…
Continued fraction expansions of Herglotz-Nevanlinna functions and generalized indefinite strings of Stieltjes type
Jonathan Eckhardt
We employ some results about continued fraction expansions of Herglotz-Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. I…
Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow
Jonathan Eckhardt, Aleksey Kostenko, Noema Nicolussi
This article is concerned with the isospectral problem \[ -f'' + \frac{1}{4} f = zωf + z^2 \upsilon f \] for the periodic conservative Camassa-Holm flow, where is a periodic re…
On the absolutely continuous spectrum of generalized indefinite strings II
Jonathan Eckhardt, Aleksey Kostenko, Teo Kukuljan
We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely…
On the absolutely continuous spectrum of generalized indefinite strings
Jonathan Eckhardt, Aleksey Kostenko
We investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous s…
The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation
Jonathan Eckhardt, Aleksey Kostenko
We solve the inverse spectral problem associated with periodic conservative multi-peakon solutions of the Camassa-Holm equation. The corresponding isospectral sets can be identifie…
The Classical Moment Problem and Generalized Indefinite Strings
Jonathan Eckhardt, Aleksey Kostenko
We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. Namely, we introduce the class of Krein-Langer strings…
The Camassa--Holm Equation and The String Density Problem
Jonathan Eckhardt, Aleksey Kostenko, Gerald Teschl
In this paper we review the recent progress in the (indefinite) string density problem and its applications to the Camassa--Holm equation.
Unique solvability of a coupling problem for entire functions
Jonathan Eckhardt
We establish the unique solvability of a coupling problem for entire functions which arises in inverse spectral theory for singular second order ordinary differential equations/two…
A Lagrangian view on complete integrability of the two-component Camassa-Holm system
Jonathan Eckhardt, Katrin Grunert
We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa-Holm system can be understood in terms of certain reparametrizations of the underlying…
Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy
Jonathan Eckhardt, Fritz Gesztesy, Helge Holden +2
We provide a construction of the two-component Camassa-Holm (CH-2) hierarchy employing a new zero-curvature formalism and identify and describe in detail the isospectral set associ…
The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data
Jonathan Eckhardt
We establish the inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data. In particular, it is employed to prove existence of weak solutions fo…
Spectral asymptotics for canonical systems
Jonathan Eckhardt, Aleksey Kostenko, Gerald Teschl
Based on continuity properties of the de Branges correspondence, we develop a new approach to study the high-energy behavior of Weyl-Titchmarsh and spectral functions of …
On Spectral Deformations and Singular Weyl Functions for One-Dimensional Dirac Operators
Alexander Beigl, Jonathan Eckhardt, Aleksey Kostenko +1
We investigate the connection between singular Weyl-Titchmarsh-Kodaira theory and the double commutation method for one-dimensional Dirac operators. In particular, we compute the s…
The inverse spectral problem for indefinite strings
Jonathan Eckhardt, Aleksey Kostenko
Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with di…
Quadratic operator pencils associated with the conservative Camassa-Holm flow
Jonathan Eckhardt, Aleksey Kostenko
We discuss direct and inverse spectral theory for a Sturm-Liouville type problem with a quadratic dependence on the eigenvalue parameter, which arises as the isospectral problem fo…
An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
Jonathan Eckhardt, Aleksey Kostenko
We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation. Utilizing the solution of the indefinite moment problem…
Inverse Spectral Problems for Schrödinger-Type Operators with Distributional Matrix-Valued Potentials
Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols +2
The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line fro…
One-dimensional Schroedinger operators with delta-prime-interactions on Cantor-type sets
Jonathan Eckhardt, Aleksey Kostenko, Mark Malamud +1
We introduce a novel approach for defining a -interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm-Liouville differential expression with m…
Singular Weyl-Titchmarsh-Kodaira theory for one-dimensional Dirac operators
Rainer Brunnhuber, Jonathan Eckhardt, Aleksey Kostenko +1
We develop singular Weyl-Titchmarsh-Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg-Mar…