Gerald Teschl

University of Vienna | UniWien · Fakultät für Mathematik

We consider essential self-adjointness on the space \(C_0^{\infty }((0,\infty ))\) of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type in \(L^2((0,\infty );dx)\). While the special case \(n=1\) is classical and it is well known that \(\tau _2(c)\big |_{C_0^{\infty }((0,\infty ))}...

We revisit the asymptotic analysis of the KdV shock problem in the soliton region. Our approach is based on the analysis of the associated Riemann–Hilbert problem and we extend the domain of validity of the asymptotic formulas while at the same time requiring less decay and smoothness for the initial data.

We study singular Sturm-Liouville operators of the form \[ \frac{1}{r_j}\left(-\frac{\mathrm d}{\mathrm dx}p_j\frac{\mathrm d}{\mathrm dx}+q_j\right),\qquad j=0,1, \] in $L^2((a,b);r_j)$, where, in contrast to the usual assumptions, the weight functions $r_j$ have different signs near the singular endpoints $a$ and $b$. In this situation the associ...

Continuous-time neural networks are a class of machine learning systems that can tackle representation learning on spatiotemporal decision-making tasks. These models are typically represented by continuous differential equations. However, their expressive power when they are deployed on computers is bottlenecked by numerical differential equation s...

The paper aims at developing the Riemann–Hilbert (RH) approach for the modified Camassa–Holm (mCH) equation on the line with non-zero boundary conditions, in the case when the solution is assumed to approach two different constants at different sides of the line. We present detailed properties of spectral functions associated with the initial data...

We develop relative oscillation theory for general Sturm-Liouville differential expressions of the form1r(−ddxpddx+q) and prove perturbation results and invariance of essential spectra in terms of the real coefficients p, q, r. The novelty here is that we also allow perturbations of the weight function r in which case the unperturbed and the pertur...

We take a closer look at the Riemann–Hilbert problem associated to one-gap solutions of the Korteweg–de Vries equation. To gain more insight, we reformulate it as a scalar Riemann–Hilbert problem on the torus. This enables us to derive deductively the model vector-valued and singular matrix-valued solutions in terms of Jacobi theta functions. We co...

We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of the Korteweg–de Vries (KdV) shock problem in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions as well as...

We develop relative oscillation theory for general Sturm-Liouville differential expressions of the form \[ \frac{1}{r}\left(-\frac{\mathrm d}{\mathrm dx} p \frac{\mathrm d}{\mathrm dx} + q\right) \] and prove perturbation results and invariance of essential spectra in terms of the real coefficients $p$, $q$, $r$. The novelty here is that we also al...

The paper aims at developing the Riemann-Hilbert (RH) approach for the modified Camassa-Holm (mCH) equation on the line with non-zero boundary conditions, in the case when the solution is assumed to approach two different constants at different sides of the line. We present detailed properties of spectral functions associated with the initial data...

We derive dispersion estimates for solutions of a one‐dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results concerning scattering for the corresponding perturbed Dirac operators which are of independent interest. Most n...

We revisit the asymptotic analysis of the KdV shock problem in the soliton region. Our approach is based on the analysis of the associated Riemann-Hilbert problem and we extend the domain of validity of the asymptotic formulas while at the same time requiring less decay and smoothness for the initial data.

We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of shock waves for the Korteweg-de Vries (KdV) equation in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions...

Continuous-depth neural models, where the derivative of the model's hidden state is defined by a neural network, have enabled strong sequential data processing capabilities. However, these models rely on advanced numerical differential equation (DE) solvers resulting in a significant overhead both in terms of computational cost and model complexity...

We take a closer look at the Riemann-Hilbert problem associated to one-gap solutions of the Korteweg-de Vries equation. To gain more insight, we reformulate it as a scalar Riemann-Hilbert problem on the torus. This enables us to derive deductively the model vector-valued and singular matrix-valued solutions in terms of Jacobi theta functions. We co...

We study perturbations of the self-adjoint periodic Sturm--Liouville operator \[ A_0 = \frac{1}{r_0}\left(-\frac{\mathrm d}{\mathrm dx} p_0 \frac{\mathrm d}{\mathrm dx} + q_0\right) \] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a fini...

We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results concerning scattering for the corresponding perturbed Dirac operators which are of independent interest. Most n...

This paper is a continuation of research started in arXiv:1406.0720, where formulas for the leading terms of the long-time asymptotics for the Toda shock wave were derived by the method of nonlinear steepest descent, in all principal domains of the space-time half plane. In the present paper we study the same Riemann--Hilbert problem in more detail...

Blood flow and ventilatory flow strongly influence the concentrations of volatile organic compounds (VOCs) in exhaled breath. The physicochemical properties of a compound (e.g., water solubility) additionally determine if the concentration of the compound in breath reflects the alveolar concentration, the concentration in the upper airways, or a mi...

This paper discusses some general aspects and techniques associated with the long-time asymptotics of steplike solutions of the Korteweg-de Vries (KdV) equation via vector Riemann--Hilbert problems. We also elaborate on an ill-posedness of the matrix Riemann-Hilbert problems for the KdV case. To the best of our knowledge this is the first time such...

The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schr\"odinger equations whose Hamiltonian is given by the generalized...

In a recent paper K.Unterkofler et al. (JBR 9 (2015), no. 3, 036002) we presented a simple two compartment model which describes the influence of inhaled concentrations on exhaled breath concentrations for volatile organic compounds (VOCs) with small Henry constants. In this paper we extend this investigation concerning the influence of inhaled c...

We prove that a solution of the Toda lattice cannot decay too fast at two different times unless it is trivial. In fact, we establish this result for the entire Toda and Kac van Moerbeke hierarchies.

In this paper we review the recent progress in the (indefinite) string density problem and its applications to the Camassa--Holm equation.

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice with steplike initial data corresponding to a rarefaction wave.

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice with steplike initial data corresponding to a rarefaction wave.

We derive a dispersion estimate for one-dimensional perturbed radial Schr\"odinger operators where the angular momentum takes the critical value $l=-\frac{1}{2}$. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.

We prove that a solution of the Toda lattice cannot decay too fast at two different times unless it is trivial. In fact, we establish this result for the entire Toda and Kac--van Moerbeke hierarchies.

We prove that a solution of the Schr\"odinger-type equation $\mathrm{i}\partial_t u= Hu$, where $H$ is a Jacobi operator with asymptotically constant coefficients, cannot decay too fast at two different times unless it is trivial.

It is shown that for a one-dimensional Schrodinger operator with a potential whose first moment is integrable the elements of the scattering matrix are in the unital Wiener algebra of functions with integrable Fourier transforms. This is then used to derive dispersion estimates for solutions of the associated Schrodinger and Klein-Gordon equations....

We derive an explicit expression for the kernel of the evolution group exp(-itH0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\exp(-\mathrm{i} t H_0)}$$\end{documen...

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.

The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schr\"odinger equations whose Hamiltonian is given by the generalized...

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.

We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case when the underlying isospectral problem has purely discrete spectrum. To this end, we introduce a natural coupling problem for entire functions, which serves as a replacement for the usual Riemann-Hilbert problem, which does not apply in th...

We develop a simple three compartment model based on mass balance equations which quantitatively describes the dynamics of breath methane concentration profiles during exercise on an ergometer. With the help of this model it is possible to estimate the endogenous production rate of methane in the large intestine by measuring breath gas concentratio...

We investigate the dependence of the $L^1\to L^\infty$ dispersive estimates for one-dimensional radial Schr\"o\-din\-ger operators on boundary conditions at $0$. In contrast to the case of additive perturbations, we show that the change of a boundary condition at zero results in the change of the dispersive decay estimates if the angular momentum i...

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 associated to all real-valued, smooth, and bounded algebro-geometric solutions of the $n$th equation of the stationary CH-2 hierarchy as the real $n$-dimensional torus $\mathbb...

We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schrodinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials. To this end we also provide new results concerning scattering for one-dimensional discrete perturbed Schrodinger operators which are of inde...

We show that the long-time behavior of solutions to the Korteweg-de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying Schr\"odinger operator) depends only on the size of the step of the initial data and on the directi...

We study the direct and inverse scattering problem for the one-dimensional Schr\"odinger equation with steplike potentials. We give necessary and sufficient conditions for the scattering data to correspond to a potential with prescribed smoothness and prescribed decay to their asymptotics. These results are important for solving the Korteweg-de Vri...

We investigate closed, symmetric $L^2(\mathbb{R}^n)$-realizations $H$ of Schr\"odinger-type operators $(- \Delta +V)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus \Sigma)}$ whose potential coefficient $V$ has a countable number of well-separated singularities on compact sets $\Sigma_j$, $j \in J$, of $n$-dimensional Lebesgue measure zero, wit...

We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Dirac equation. To this end we develop basic scattering theory and establish a limiting absorption principle for discrete perturbed Dirac operators.

We show that for a Jacobi operator with coefficients whose (j+1)'th moments are summable the j'th derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known dispersive estimates with integrable time decay for the time dependent Jacobi equation in the resonant...

In this paper we develop a simple two compartment model which extends the Farhi equation to the case when the inhaled concentration of a volatile organic compound (VOC) is not zero. The model connects the exhaled breath concentration of VOCs with physiological parameters such as endogenous production rates and metabolic rates. Its validity is teste...

We show that for a one-dimensional Schr\"odinger operator with a potential whose (j+1)'th moment is integrable the j'th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schr\"od...

We derive a dispersion estimate for one-dimensional perturbed radial Schr\"odinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.

In this paper we develop a simple two compartment model which extends the Farhi equation to the case when the inhaled concentration of a volatile organic compound (VOC) is not zero. The model connects the exhaled breath concentration of systemic VOCs with physiological parameters such as endogenous production rates and metabolic rates. Its validi...

Volatile organic compounds emitted by a human body form a chemical signature capable of providing invaluable information on the physiological status of an individual and, thereby, could serve as signs-of-life for detecting victims after natural or man-made disasters. In this review a database of potential biomarkers of human presence was created on...

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 singular Weyl function of the commuted operator in terms of the data from the original operator. These results are then applied to radial Dirac operators in order to show...

We derive an asymptotic expansion for the Weyl function of a one-dimensional Schr\"odinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of distributions.

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 $2\times2$ first order canonical systems. Our results improve several classical results and solve open problems posed by previous authors. Furthermore, they are applied to radial D...

We show that for a one-dimensional Schr\"odinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive dispersion estimates for solutions of the associated Schr\"odinger and Klein-Gordon equations. In particular, we...

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-Marchenko and Hochstadt-Liebermann type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.

We introduce a novel approach for defining a δ′-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm–Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with δ′...

We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to free backgrounds. The middle region, where...

We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"odinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials.

Wir betrachten eine endliche oder unendliche Grundgesamtheit (z. B. die Bevölkerung eines Landes, alle Artikel aus einer laufenden Produktion). Wenn sie endlich ist, so bezeichnen wir ihren Umfang mit N. An den Elementen der Grundgesamtheit interessiert uns ein Merkmal X (z. B. Alter, Durchmesser, … ).

Bis jetzt haben wir es fast ausschließlich mit Funktionen einer Variable zu tun gehabt. Nicht in jeder Situation kommt man aber damit aus. So wird z. B. der Ertrag einer Firma im Allgemeinen von mehreren Faktoren abhängen und ist somit eine Funktion von mehreren Variablen. Diesen Fall wollen wir nun eingehender untersuchen.

Betrachten wir die Funktion \( f(x) = \frac{{\sin (x)}}{x}. \) Sie ist an der Stelle \( {x_0} = 0 \) nicht definiert, es gibt hier also keinen Funktionswert. Wir können uns nun fragen, wie sich die Funktionswerte verhalten, wenn sich das Argument \( x \) dem Wert 0 nähert. Man könnte vermuten, dass die Funktionswerte dann wegen des Faktors \( \frac...

Polynome und rationale Funktionen haben die angenehme Eigenschaft, dass man ihre Funktionswerte leicht, nämlich nur unter Verwendung der Grundrechenoperationen +, -, ., /, berechnen kann. Das sind aber die einzigen Operationen, die ein Computer von sich aus beherrscht! Mehr muss er aber zum Glück auch nicht können, denn alle komplizierteren Funktio...

Angenommen, \( x(t) \) beschreibt die Größe einer Population (z. B. Bakterien) zur Zeit \( t \). Im einfachsten Fall ist die Zunahme der Population proportional zur vorhandenen Population, d.h.,$$ \frac{d}{{dt}}x(t) = \mu x(t)

Wir haben in Abschnitt 19.2 gesehen, dass eine Funktion \( f \) in der Nähe einer Stelle \( {x_0} \) durch ihre Tangente angenähert werden kann. Oft ist diese Approximation durch ein Polynom vom Grad 1 aber nicht gut genug. Ist es möglich dieses Verfahren zu verfeinern, indem man Polynome höheren Grades verwendet? Diese Frage führt uns zu den so ge...

Nehmen wir an, wir haben ein akustisches Signal, das wir auf einem Computer abspeichern möchten. Das Signal könnte wie in Abbildung 22.1 aussehen.

Eine grundlegende Aufgabe der Statistik besteht darin, Informationen über bestimmte Objekte zu gewinnen, ohne dass dabei alle Objekte untersucht werden müssen. Es werden also Daten über eine Stichprobe erhoben und in der Folge ausgewertet, um daraus Schlussfolgerungen ziehen zu können.

Im letzten Kapitelhaben wir Funktionen differenziert.

Die Normalverteilung ist ohne Zweifel die wichtigste Verteilung der Statistik. Sie ist zu erwarten, wenn ein Merkmal, zum Beispiel die Füllmenge X von automatisch abgefüllten Gläsern, sich aus einer Summe von vielen zufälligen, unabhängigen Einflüssen, von denen keiner dominierend ist, zusammensetzt. Solche Einflüsse können zum Beispiel sein: Auswi...

Wenn Sie eine Münze werfen, so bestimmt der „Zufall“, ob das Ergebnis „Kopf“ oder „Zahl“ sein wird. Wenn aus einer Warenlieferung 100 Glühbirnen zufällig entnommen werden, so kann man ebenfalls nicht vorhersagen, wie groß die Anzahl der defekten Glühbirnen darunter sein wird.

Wir besprechen in diesem Kapiteleinige in der Praxis wichtige diskrete Verteilungen: die hypergeometrische Verteilung, die Binomialverteilung und die PoissonVerteilung. Im vorhergehenden Kapitelsind Ihnen (ohne dass sie so genannt wurden) bereits hypergeometrisch und binomialverteilte Zufallsvariable begegnet. Da sie so oft auftreten, überlegt man...

The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schr\"odinger-type operators on a half-line from the underlying Weyl-Titchmarsh function.

We investigate the connections between Weyl-Titchmarsh-Kodaira theory for one-dimensional Schr\"odinger operators and the theory of $n$-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schr\"odinger operator to be $n$-entire in terms of square integrability of derivatives (w.r.t. the spectral p...

We provide an abstract framework for singular one-dimensional Schr̈odinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show...

Erinnern Sie sich an die Division mit Rest aus Satz 2.49: Wenn \( a \in \mathbb{Z}\ \) und \( m \in \mathbb{N}\ \), so kann man \( a \) in der Form $$ a = q \cdot m + r\

Graphen werden oft zur Modellierung von Transportproblemen verwendet. Transportiert wird zum Beispiel Wasser oder Öl in einem Leitungsnetz. Allgemeiner können wir aber auch von einem Fluss von Information, Emails, Anrufen durch ein Kommunikationsnetz, von Menschen im öffentlichen Verkehrsnetz, von PKWs oder Warenlieferungen durch ein Straßennetz, u...

Erinnern Sie sich an die Definition eines Körpers in Abschnitt 3.2. Das ist eine Menge \( \mathbb{K} \) gemeinsam mit zwei Verknüpfungen, Addition und Multiplikation genannt, die bestimmte Eigenschaften erfüllen. Insbesondere gibt es für jedes Element \( a \) des Körpers ein inverses Element \( -a \) bezüglich der Addition und für jedes \( a \ne 0...

In den meisten Programmiersprachen steht der Datentyp array (engl. für Feld, Anordnung) zur Verfügung. In einem array können mehrere gleichartige Elemente zusammengefasst werden, auf die mithilfe von Indizes zugegriffen wird. Hat jedes Element einen Index, so entspricht der array einem Vektor; wird jedes Element durch zwei Indizes angegeben, so füh...

In vielen Problemen der Praxis werden Lösungen gesucht, die bestimmten Einschränkungen genügen. Diese Einschränkungen können oft durch lineare Ungleichungen beschrieben werden.

Relationen sind ein mathematisches Hilfsmittel, um Beziehungen zwischen einzelnen Objekten zu beschreiben. Sie werden zum Beispiel in relationalen Datenbanken und in der theoretischen Informatik (z. B. formale Sprachen) verwendet.

Bäume gehören zu den wichtigsten Typen von Graphen. Sie sind grundlegende Bausteine für alle Graphen. Darüber hinaus sind sie gut geeignet zur Darstellung von Strukturen bzw. Abläufen (z. B. Suchen, Sortieren).

Die Kombinatorik untersucht die verschiedenen Möglichkeiten, Objekte anzuordnen bzw. auszuwählen. Sie ist im 17. Jahrhundert durch Fragestellungen begründet worden, die durch Glücksspiele aufgekommen sind. Viele Abzählprobleme können formuliert werden, indem man geordnete oder ungeordnete Auswahlen von Objekten trifft, die Permutationen bzw.

Graphen werden in vielen Anwendungsgebieten, wie zum Beispiel Informations- und Kommunikationstechnologien, Routenplanung oder Projektplanung eingesetzt. Sie helfen bei der Beantwortung von Fragen wie: Auf welchem Weg können Nachrichten im Internet möglichst effizient vom Sender zum Empfänger geleitet werden? Wo sollen neue Straßen gebaut werden, u...

Größen wie Geschwindigkeit, Kraft, usw. sind dadurch gekennzeichnet, dass sie nicht nur einen Betrag, sondern auch eine Richtung haben. Sie können durch Pfeile veranschaulicht werden. Man nennt solche Größen auch vektorielle Größen, im Unterschied zu so genannten skalaren Größen, wie etwa einer Temperatur, die durch eine einzige reelle Zahl (einen...

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data.