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Christian Remling

Publications and source records attributed to Christian Remling.

At least 19 recordsLinked to original sources
arXiv2025

Polynomial Toda maps are transfer matrices

We consider entire matrix functions $A(z)$ taking values in $\operatorname{SL}(2,\mathbb C)$. These map pairs of Herglotz functions by acting pointwise as linear fractional transformations. The main examples of such Toda maps are provided by transfer matrices of differential and difference operators and by the cocycles associated with the classical integrable systems (Toda, KdV, etc.) on these operators. Here we consider polynomial matrix functions $A(z)$. We describe these in terms of a factorization, and we then prove that if $A$ induces a Toda map, then $A$ is essentially a transfer matrix.

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arXiv2024

Reflectionless operators and automorphic Herglotz functions

I am interested in canonical systems and Dirac operators that are reflectionless on an open set. In this situation, the half line $m$ functions are holomorphic continuations of each other and may be combined into a single function. By passing to the universal cover of its domain, we then obtain a one-to-one correspondence of these operators with Herglotz functions that are automorphic with respect to the Fuchsian group of covering transformations. I investigate the properties of this formalism, with particular emphasis given to the measures that are automorphic in a corresponding sense. This will shed light on the reflectionless operators as a topological space, on their extreme points, and on how the heavily studied smaller space of finite gap operators sits inside the (much) larger space.

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arXiv2024

Reflectionless Dirac operators and canonical systems

We study canonical systems that are reflectionless on an open set. In this situation, the two half line $m$ functions are holomorphic continuations of each other and may thus be combined into a single holomorphic function. This idea was explored in [11], and we continue these investigations here. We focus on Dirac operators and especially their interplay with canonical systems, and we provide a more general and abstract framework.

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arXiv2019

The essential spectrum of canonical systems

We study the minimum of the essential spectrum of canonical systems $Ju'=-zHu$. Our results can be described as a generalized and more quantitative version of the characterization of systems with purely discrete spectrum, which was recently obtained by Romanov and Woracek [6]. Our key tool is oscillation theory.

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arXiv2018

Oscillation theory and semibounded canonical systems

Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems $Ju'=-zHu$ and then use it to discuss semibounded operators from this point of view. Our main new result is a characterization of systems with purely discrete spectrum in terms of the asymptotics of their coefficient functions; we also discuss the exponential types of the transfer matrices.

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arXiv2018

Generalized Toda flows

The classical hierarchy of Toda flows can be thought of as an action of the (abelian) group of polynomials on Jacobi matrices. We present a generalization of this to the larger groups of $C^2$ and entire functions, and in this second case, we also introduce associated cocycles and in fact give center stage to this object.

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arXiv2017

Toda maps, cocycles, and canonical systems

I present a discussion of the hierarchy of Toda flows that gives center stage to the associated cocycles and the maps they induce on the $m$ functions. In the second part, these ideas are then applied to canonical systems; an important feature of this discussion will be my proposal that the role of the shift on Jacobi matrices should now be taken over by the more general class of twisted shifts.

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arXiv2014

Generalized reflection coefficients

I consider general reflection coefficients for arbitrary one-dimensional whole line differential or difference operators of order $2$. These reflection coefficients are semicontinuous functions of the operator: their absolute value can only go down when limits are taken. This implies a corresponding semicontinuity result for the absolutely continuous spectrum, which applies to a very large class of maps. In particular, we can consider shift maps (thus recovering and generalizing a result of Last-Simon) and flows of the Toda and KdV hierarchies (this is new). Finally, I evaluate an attempt at finding a similar general setup that gives the much stronger conclusion of reflectionless limit operators in more specialized situations.

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arXiv2011

Ergodic Jacobi matrices and conformal maps

We study structural properties of the Lyapunov exponent $γ$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-γ+iπk$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem.

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arXiv2010

The absolutely continuous spectrum of Jacobi matrices

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case. The main theme is the following: It is extremely difficult to produce absolutely continuous spectrum in one space dimension and thus its existence has strong implications.

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arXiv2010

Uniqueness of reflectionless Jacobi matrices and the Denisov-Rakhmanov Theorem

If a Jacobi matrix $J$ is reflectionless on $(-2,2)$ and has a single $a_{n_0}$ equal to 1, then $J$ is the free Jacobi matrix $a_n\equiv 1$, $b_n\equiv 0$. I'll discuss this result and its generalization to arbitrary sets and present several applications, including the following: if a Jacobi matrix has some portion of its $a_n$'s close to 1, then one assumption in the Denisov-Rakhmanov Theorem can be dropped.

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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?

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arXiv2008

Reflectionless Herglotz functions and generalized Lyapunov exponents

We study several related aspects of reflectionless Jacobi matrices. Our first set of results deals with the singular part of reflectionless measures. We then introduce and discuss Lyapunov exponents, density of states measures, and other related quantities in a \textit{general} setting. This is related to the previous material because the density of states measures are reflectionless on certain sets.

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arXiv2008

The absolutely continuous spectrum of one-dimensional Schr"odinger operators

This paper deals with general structural properties of one-dimensional Schr"odinger operators with some absolutely continuous spectrum. The basic result says that the omega limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.

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