arXiv subjects

Rudi Weikard

Publications and source records attributed to Rudi Weikard.

At least 19 recordsLinked to original sources

arXiv2026

Self-adjoint extensions of symmetric relations associated with systems of ordinary differential equations with distributional coefficients

We study the extension theory for the two-dimensional first-order system $Ju' +qu = wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are real distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we characterize the boundary conditions for solutions $u$ in the closure of the minimal relation, as well as describe the properties of quasi-boundary conditions which yield self-adjoint extensions. We then apply these ideas to a popular extension of non-negative minimal relations: the Krein-von Neumann extension. For more context on how the Krein-von Neumann is defined, an appendix shows a construction of the Friedrichs extension from which the Krein-von Neumann is traditionally defined.

math.SP↗

arXiv2025

Bright and dark breathers on an elliptic wave in the defocusing mKdV equation

Breathers on an elliptic wave background consist of nonlinear superpositions of a soliton and a periodic wave, both traveling with different wave speeds and interacting periodically in the space-time. For the defocusing modified Korteweg-de Vries (mKdV) equation, the construction of general breathers has been an open problem since the elliptic wave is related to the elliptic degeneration of the hyperelliptic solutions of genus two. We have found the new representation of eigenfunctions of the Lax operator associated with the elliptic wave, which enables us to solve this open problem and to construct two families of breathers with bright (elevation) and dark (depression) profiles.

nlin.SI↗

arXiv2023

The limit-point/limit-circle classification for ordinary differential equations with distributional coefficients

We investigate the limit-point/limit-circle classification for the differential equation $Ju'+qu=λwu$ where $J=\big(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\big)$ and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. We identify the situations when the classical alternative of Weyl works and when it fails.

math.SP↗

arXiv2023

On Fourier expansions for systems of ordinary differential equations with distributional coefficients

We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we construct a generalized Weyl-Titchmarsh $m$-function with corresponding spectral measure $τ$ and a generalized Fourier transform after imposing certain conditions on $J$, $q$, and $w$. Different conditions are motivated and studied in the later sections. A Fatou-type identity needed for our result is recorded in the appendix.

math.SP↗

arXiv2023

On the spectral theory of systems of first order equations with periodic distributional coefficients

We establish a Floquet theorem for a first-order system of differential equations $u'=ru$ where $r$ is an $n\times n$-matrix whose entries are periodic distributions of order $0$. Then we investigate, when $n=1$ and $n=2$, the spectral theory for the equation $Ju'+qu=wf$ on $\mathbb R$ when $J$ is a real, constant, invertible, skew-symmetric matrix and $q$ and $w$ are periodic matrices whose entries are real distributions of order $0$ with $q$ symmetric and $w$ non-negative.

math.SP↗

arXiv2023

Green's functions for first-order systems of ordinary differential equations without the unique continuation property

This paper is a contribution to the spectral theory associated with the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with $Ju'+qu=wf$ and show the existence of Green's functions for self-adjoint relations even if unique continuation of solutions fails.

math.SP↗

arXiv2023

On the spectral theory for first-order systems without the unique continuation property

We consider the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. In this situation it may happen that there is no existence and uniqueness theorem for balanced solutions of a given initial value problem. We describe the set of solutions the equation does have and establish that the adjoint of the minimal operator is still the maximal operator, even though unique continuation of balanced solutions fails.

math.CA↗

arXiv2019

Sign-Changing Points of Solutions of Homogeneous Sturm-Liouville Equations with Measure-Valued Coefficients

In this paper we investigate sign-changing points of nontrivial real-valued solutions of homogeneous Sturm-Liouville differential equations of the form $-d(du/dα)+udβ=0$, where $dα$ is a positive Borel measure supported everywhere on $(a,b)$ and $dβ$ is a locally finite real Borel measure on $(a,b)$. Since solutions for such equations are functions of locally bounded variation, sign-changing points are the natural generalization of zeros. We prove that sign-changing points for each nontrivial real-valued solution are isolated in $(a,b)$. We also prove a Sturm-type separation theorem for two nontrivial linearly independent solutions, and conclude the paper by proving a Sturm-type comparison theorem for two differential equations with distinct potentials.

math.CA↗

arXiv2019

Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients

We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Also, we do not pose the definiteness condition customarily required for the coefficients of the equation. Specifically, we construct minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we determine Green's function and prove the existence of a spectral (or generalized Fourier) transformation. We have a closer look at the special cases when the endpoints of the interval $(a,b)$ are regular as well as the case of a $2\times2$ system. Two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations.

math.CA↗

arXiv2017

On Leighton's Comparison Theorem

We give a simple proof of a fairly flexible comparison theorem for equations of the type $-(p(u'+su))'+rp(u'+su)+qu=0$ on a finite interval where $1/p$, $r$, $s$, and $q$ are real and integrable. Flexibility is provided by two functions which may be chosen freely (within limits) according to the situation at hand. We illustrate this by presenting some examples and special cases which include Schrödinger equations with distributional potentials as well as Jacobi difference equations.

math.CA↗

arXiv2015

Donoghue-Type $m$-Functions for Schrödinger Operators with Operator-Valued Potentials

Given a complex, separable Hilbert space $\mathcal{H}$, we consider self-adjoint $L^2$-realizations of differential expressions $τ= - (d^2/dx^2) I_{\mathcal{H}} + V(x)$, on half-lines and on the real line (assuming the limit-point property of $τ$ at $\pm \infty$). Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in \mathcal{B}(\mathcal{H})$ such that $V(\cdot)$ is weakly measurable, the operator norm $\|V(\cdot)\|_{\mathcal{B}(\mathcal{H})}$ is locally integrable, and $V(\cdot) = V(\cdot)^*$ a.e. In a nutshell, a Donoghue-type $m$-function $M_{A,\mathcal{N}_i}^{Do}(\cdot)$ associated with self-adjoint extensions $A$ of a closed, symmetric operator $\dot A$ in $\mathcal{H}$ with deficiency spaces $\mathcal{N}_z = \ker \big({\dot A}^* - z I_{\mathcal{H}}\big)$ and corresponding orthogonal projections $P_{\mathcal{N}_z}$ onto $\mathcal{N}_z$ is given by $$ M_{A,\mathcal{N}_i}^{Do}(z) = zI_{\mathcal{N}_i} + (z^2+1) P_{\mathcal{N}_i} (A - z I_{\mathcal{H}})^{-1} P_{\mathcal{N}_i}\big\vert_{\mathcal{N}_i} \,, \quad {\rm Im}(z)\neq 0. $$ For half-line and full-line Schrödinger operators, the role of $\dot A$ is played by a suitably defined minimal Schrödinger operator which will be shown to be completely non-self-adjoint. The latter property is used to prove that the corresponding operator-valued measures in the Herglotz--Nevanlinna representations of the Donoghue-type $m$-functions corresponding to self-adjoint half-line and full-line Schrödinger operators encode the entire spectral information of the latter.

math.SP↗

arXiv2013

Some Remarks on the Spectral Problem Underlying the Camassa-Holm Hierarchy

We consider left-definite eigenvalue problems $A ψ= λB ψ$, with $A \geq \varepsilon I$ for some $\varepsilon > 0$ and $B$ self-adjoint, but $B$ not necessarily positive or negative definite, applicable, in particular, to the eigenvalue problem underlying the Camassa-Holm hierarchy. In fact, we will treat a more general version where $A$ represents a positive definite Schrödinger or Sturm-Liouville operator $T$ in $L^2(\bbR; dx)$ associated with a differential expression of the form $τ= - (d/dx) p(x) (d/dx) + q(x)$, $x \in \bbR$, and $B$ represents an operator of multiplication by $r(x)$ in $L^2(\bbR; dx)$, which, in general, is not a weight, that is, it is not nonnegative a.e.\ on $\bbR$. Our methods naturally permit us to treat certain classes of distributions (resp., measures) for the coefficients $q$ and $r$ and hence considerably extend the scope of this (generalized) eigenvalue problem, without having to change the underlying Hilbert space $L^2(\bbR; dx)$. Our approach relies on rewriting the eigenvalue problem $A ψ= λB ψ$ in the form $A^{-1/2} B A^{-1/2} χ= λ^{-1} χ$, $χ= A^{1/2} ψ$, and a careful study of (appropriate realizations of) the operator $A^{-1/2} B A^{-1/2}$ in $L^2(\bbR; dx)$. In the course of our treatment we employ a supersymmetric formalism which permits us to factor the second-order operator $T$ into a product of two first-order operators familiar from (and inspired by) Miura's transformation linking the KdV and mKdV hierarchy of nonlinear evolution equations. We also treat the case of periodic coefficients $q$ and $r$, where $q$ may be a distribution and $r$ generates a measure and hence no smoothness is assumed for $q$ and $r$.

math.SP↗

arXiv2013

On Spectral Theory for Schrödinger Operators with Operator-Valued Potentials

Given a complex, separable Hilbert space $\cH$, we consider differential expressions of the type $τ= - (d^2/dx^2) + V(x)$, with $x \in (a,\infty)$ or $x \in \bbR$. Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in \cB(\cH)$ such that $V(\cdot)$ is weakly measurable and the operator norm $\|V(\cdot)\|_{\cB(\cH)}$ is locally integrable. We consider self-adjoint half-line $L^2$-realizations $H_α$ in $L^2((a,\infty); dx; \cH)$ associated with $τ$, assuming $a$ to be a regular endpoint necessitating a boundary condition of the type $\sin(α)u'(a) + \cos(α)u(a)=0$, indexed by the self-adjoint operator $α= α^* \in \cB(\cH)$. In addition, we study self-adjoint full-line $L^2$-realizations $H$ of $τ$ in $L^2(\bbR; dx; \cH)$. In either case we treat in detail basic spectral theory associated with $H_α$ and $H$, including Weyl--Titchmarsh theory, Green's function structure, eigenfunction expansions, diagonalization, and a version of the spectral theorem.

math.SP↗

arXiv2013

Stability for the inverse resonance problem for the CMV operator

For the class of unitary CMV operators with super-exponentially decaying Verblunsky coefficients we give a new proof of the inverse resonance problem of reconstructing the operator from its resonances - the zeros of the Jost function. We establish a stability result for the inverse resonance problem that shows continuous dependence of the operator coefficients on the location of the resonances.

math.SP↗

arXiv2011

On a class of Model Hilbert Spaces

We provide a detailed description of the model Hilbert space $L^2(\bbR; dΣ; \cK)$, were $\cK$ represents a complex, separable Hilbert space, and $Σ$ denotes a bounded operator-valued measure. In particular, we show that several alternative approaches to such a construction in the literature are equivalent. These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.

math.SP↗

arXiv2011

Initial Value Problems and Weyl--Titchmarsh Theory for Schrödinger Operators with Operator-Valued Potentials

We develop Weyl-Titchmarsh theory for self-adjoint Schrödinger operators $H_α$ in $L^2((a,b);dx;\cH)$ associated with the operator-valued differential expression $τ=-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $α= α^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(α)u'(a)+\cos(α)u(a)=0, $$ with $u$ lying in the domain of the underlying maximal operator $H_{\max}$ in $L^2((a,b);dx;\cH)$ associated with $τ$. More precisely, we establish the existence of the Weyl-Titchmarsh solution of $H_α$, the corresponding Weyl-Titchmarsh $m$-function $m_α$ and its Herglotz property, and determine the structure of the Green's function of $H_α$. Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient $V$, -y" + (V - z) y = f \, \text{on} \, (a,b), y(x_0) = h_0, \; y'(x_0) = h_1, under the following general assumptions: $(a,b)\subseteq\bbR$ is a finite or infinite interval, $x_0\in(a,b)$, $z\in\bbC$, $V:(a,b)\to\cB(\cH)$ is a weakly measurable operator-valued function with $\|V(\cdot)\|_{\cB(\cH)}\in L^1_\loc((a,b);dx)$, and $f\in L^1_{\loc}((a,b);dx;\cH)$, with $\cH$ a complex, separable Hilbert space. We also study the analog of this initial value problem with $y$ and $f$ replaced by operator-valued functions $Y, F \in \cB(\cH)$. Our hypotheses on the local behavior of $V$ appear to be the most general ones to date.

math.SP↗

arXiv2009

On the Inverse Resonance Problem for Schrodinger Operators

We consider Schrödinger operators on [0,\infty) with compactly supported, possibly complex-valued potentials in L^1([0,\infty)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.

math-ph↗

arXiv2003

An Explicit Characterization of Calogero--Moser Systems

Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Calogero--Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple collisions between the poles of solutions.

nlin.SI↗