# Christian SeifertTechnische Universität Hamburg | TUHH · Institute of Mathematics

Christian Seifert

Dr. rer. nat.

## About

97

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Introduction

Additional affiliations

April 2021 - August 2021

October 2019 - March 2020

October 2015 - August 2016

## Publications

Publications (97)

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed...

We study abstract sufficient criteria for cost-uniform open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates...

We consider time-dependent Desch–Schappacher perturbations of non-autonomous abstract Cauchy problems and apply our result to non-autonomous uniformly strongly elliptic differential operators on \(\mathrm {L}^p\)-spaces.

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time T>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlen...

Spectral properties of Schr\"odinger operators on compact metric graphs are studied and special emphasis is put on differences in the spectral behavior between different classes of vertex conditions. We survey recent results especially for $\delta$ and $\delta'$ couplings and demonstrate the spectral properties on many examples. Amongst other thing...

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed...

Given a Banach space X and an additional coarser Hausdorff locally convex topology τ on X, we characterise the generators of τ-bi-continuous semigroups in the spirit of the Lumer–Phillips theorem, i.e. by means of dissipativity with respect to a directed system of seminorms and a range condition.

Lehrveranstaltungen mit sehr großen Teilnehmerzahlen erzeugen ein entsprechend hohes Prüfungsaufkommen. Die elektronische Durchführung solcher Prüfungen inklusive einer automatisierten Korrektur und Bewertung kann zum einen den Arbeitsaufwand erheblich reduzieren und liefert zum anderen eine neue Dimension möglicher Prüfungsfragen. Außerdem lässt s...

Given a Banach space $X$ and an additional coarser Hausdorff locally convex topology $\tau$ on $X$ we characterise the generators of $\tau$-bi-continuous semigroups in the spirit of the Lumer--Phillips theorem, i.e. by means of dissipativity w.r.t. a directed system of seminorms and a range condition. This is the preprint of the corresponding artic...

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time T>0 by taking into account the orbit of the initial value under the semigroup for t in [0,T], measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation...

We study non-autonomous observation systems \begin{align*} \dot{x}(t) = A(t) x(t),\quad y(t) = C(t) x(t),\quad x(0) = x_0\in X, \end{align*} where $(A(t))$ is a strongly measurable family of closed operators on a Banach space $X$ and $(C(t))$ is a family of bounded observation operators from $X$ to a Banach space $Y$. Based on an abstract uncertain...

This paper generalizes the abstract method of proving an observability estimate by combining an uncertainty principle and a dissipation estimate. In these estimates we allow for a large class of growth/decay rates satisfying an integrability condition. In contrast to previous results, we use an iterative argument which enables us to give an asympto...

This paper is dedicated to e-assessment in mathematics and the development of the corresponding electronic tasks. Due to its electronic nature e-assessment can be used to efficiently provide a huge number of tasks for training. This makes it a valuable tool for large classes, which are typical for mathematics as a service subject. In order to devel...

We study (cost-uniform approximate) null-controllability of parabolic equations in \begin{document}$ L_p( \mathbb{R}^d) $\end{document} and provide explicit bounds on the control cost. In particular, we consider systems of the form \begin{document}$ \dot{x}(t) = -A_px(t) + {\bf{1}}_ E u(t) $\end{document}, \begin{document}$ x(0) = x_0\in L_p ( \mat...

We consider time-dependent Desch-Schappacher perturbations of non-autonomous abstract Cauchy problems and apply our result to non-autonomous uniformly strongly elliptic differential operators on $\mathrm{L}^p$-spaces.

This chapter is devoted to the study of evolutionary inclusions. In contrast to evolutionary equations, we will replace the skew-selfadjoint operator A by a so-called maximal monotone relation A ⊆ H × H in the Hilbert space H. The resulting problem is then no longer an equation, but just an inclusion; that is, we consider problems of the form (u,f)...

This chapter is concerned with the study of problems of the form ∂t,νMn(∂t,ν)+AUn=F for a suitable sequence of material laws Mnn when A ≠ 0. The aim of this chapter will be to provide the conditions required for convergence of the material law sequence to imply the existence of a limit material law M such that the limit U =limn→∞Un exists and satis...

In this chapter, we discuss a first application of the time derivative operator constructed in the previous chapter. More precisely, we analyse well-posedness of ordinary differential equations and will at the same time provide a Hilbert space proof of the classical Picard–Lindelöf theorem (There are different notions for this theorem. It is also c...

The power of the functional analytic framework for evolutionary equations lies in its variety. In fact, as we have outlined in earlier chapters, it is possible to formulate many differential equations in the form ∂tM(∂t)+AU=F.

Let H be a Hilbert space and ν∈ℝ. We saw in the previous chapter how initial value problems can be formulated within the framework of evolutionary equations. More precisely, we have studied problems of the form ∂t,νM0+M1+AU=0on0,∞,M0U(0+)=M0U0 for U0 ∈ H, M0, M1 ∈ L(H) and A:dom(A)⊆H→H skew-selfadjoint; that is, we have considered material laws of...

In this chapter we turn our focus back to causal operators. In Chap. 5 we found out that material laws provide a class of causal and autonomous bounded operators. In this chapter we will present another proof of this fact, which rests on a result which characterises functions in L2(ℝ;H) with support contained in the non-negative reals; the celebrat...

It is the aim of this chapter to define a derivative operator on a suitable L2-space, which will be used as the derivative with respect to the temporal variable in our applications. As we want to deal with Hilbert space-valued functions, we start by introducing the concept of Bochner–Lebesgue spaces, which generalises the classical scalar-valued Lp...

We will gather some information on operators in Banach and Hilbert spaces. Throughout this chapter let X0, X1, and X2 be Banach spaces and H0, H1, and H2 be Hilbert spaces over the field 𝕂∈{ℝ,ℂ}.

In this chapter, we shall discuss and present the first major result of the manuscript: Picard’s theorem on the solution theory for evolutionary equations which is the main result of Picard (A structural observation for linear material laws in classical mathematical physics. In Mathematical Methods in the Applied Sciences, vol 32, 2009, pp 1768–180...

In this chapter we introduce the Fourier–Laplace transformation and use it to define operator-valued functions of ∂t,ν; the so-called material law operators. These operators will play a crucial role when we deal with partial differential equations. In the equations of classical mathematical physics, like the heat equation, wave equation or Maxwell’...

This chapter is devoted to the study of inhomogeneous boundary value problems. For this, we shall reformulate the boundary value problem again into a form which fits within the general framework of evolutionary equations. In order to have an idea of the type of boundary values which make sense to study, we start off with a section that deals with t...

This chapter is devoted to a small tour through a variety of evolutionary equations. More precisely, we shall look into the equations of poro-elastic media, (time-)fractional elasticity, thermodynamic media with delay as well as visco-elastic media. The discussion of these examples will be similar to that of the examples in the previous chapter in...

In this chapter, we address the issue of maximal regularity. More precisely, we provide a criterion on the ‘structure’ of the evolutionary equation ∂t,νM(∂t,ν)+A¯U=F in question and the right-hand side F in order to obtain U∈dom(∂t,νM(∂t,ν))∩dom(A). If F∈L2,ν(ℝ;H), U∈dom(∂t,νM(∂t,ν))∩dom(A) is the optimal regularity one could hope for. However, one...

In this chapter we study the exponential stability of evolutionary equations. Roughly speaking, exponential stability of a well-posed evolutionary equation ∂t,νM(∂t,ν)+AU=F means that exponentially decaying right-hand sides F lead to exponentially decaying solutions U. The main problem in defining the notion of exponential decay for a solution of a...

Up until now we have dealt with evolutionary equations of the form (∂t,νM(∂t,ν)+A¯)U=F for some given F∈L2,ν(ℝ;H) for some Hilbert space H, a skew-selfadjoint operator A in H and a material law M defined on a suitable half-plane satisfying an appropriate positive definiteness condition with ν∈ℝ chosen suitably large. Under these conditions, we esta...

Previously, we focussed on evolutionary equations of the form ∂t,νM(∂t,ν)+A¯U=F. In this chapter, where we turn back to well-posedness issues, we replace the material law operator M(∂t,ν), which is invariant under translations in time, by an operator of the form ℳ+∂t,ν−1N, where both ℳ and N are bounded linear operators in L2,ν(ℝ;H). Thus, it is th...

We study abstract sufficient criteria for open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only nee...

This paper is dedicated to e-assessment in mathematics and the development of the
corresponding electronic tasks. Due to its electronic nature e-assessment can be used to
efficiently provide a huge number of tasks for training. This makes it a valuable tool for large classes, which are typical for mathematics as a service subject. In order to devel...

We consider operators A on a sequentially complete Hausdorff locally convex space X such that -A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-A$$\end{document} gener...

Ein wesentlicher Aspekt einer Lehrveranstaltung zur Ingenieurmathematik ist der konkrete Bezug der mathematischen Inhalte zu den persönlichen Anwendungsfächern – wie Elektrotechnik oder Mechanik – der Studierenden. In großen Kursen, in denen unterschiedliche Studiengänge gleichzeitig unterrichtet werden, lässt sich dies mittels Übungsaufgaben in pe...

We study the influence of certain geometric perturbations on the spectra of self-adjoint Schrödinger operators on compact metric graphs. Results are obtained for permutation invariant vertex conditions, which, amongst others, include δ and δ′-type conditions. We show that adding edges to the graph or joining vertices changes the eigenvalues monoton...

We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e., of the harmonic extension...

We study null-controllability of parabolic equations in Banach spaces. We show that a generalized uncertainty principle and a dissipation estimate imply (approximate) null-controllability. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, we do not assume reflexivity of the under...

We study band-dominated operators on (subspaces of) Lp-spaces over metric measure spaces of bounded geometry satisfying an additional property. We single out core assumptions to obtain, in an abstract setting, definitions of limit operators, characterizations of compactness and Fredholmness using limit operators; and thus also spectral consequences...

This is the final version of the lecture notes of the 23rd Internet Seminar on Evolutionary Equations, see also https://www.mat.tuhh.de/isem23/.

For the study of electromagnetic waves in a 2D environment, the Helmholtz equation can be reformulated in boundary integral form by the so‐called Contour Integral Method (CIM). The influence of uncertainty in the geometry is studied via Polynomial Chaos Expansion (PCE). A distinction is made between the so‐called intrusive and non‐intrusive PCE. Bo...

We show to what extend fractional powers of non‐densely defined sectorial operators on Banach spaces can still be described by the harmonic extension approach.

We study band-dominated operators on (subspaces of) $L_p$-spaces over metric measure spaces of bounded geometry satisfying an additional property. We single out core assumptions to obtain, in an abstract setting, definitions of limit operators, characterizations of compactness and Fredholmness using limit operators; and thus also spectral consequen...

This paper presents a hybrid boundary element method for the efficient simulation of substrate integrated waveguide (SIW) horn antennas. It is applicable with good accuracy to relatively thin structures with conventional circular ground vias. In the multi-scale simulations, a 2D contour integral method is used for the modeling of the fields inside...

Let $X,Y$ be Banach spaces, $(S_t)_{t \geq 0}$ a $C_0$-semigroup on $X$, $-A$ the corresponding infinitesimal generator on $X$, $C$ a bounded linear operator from $X$ to $Y$, and $T > 0$. We consider the system \[ \dot{x}(t) = -Ax(t), \quad y(t) = Cx(t) \quad t\in (0,T], \quad x(0) = x_0 \in X. \] We provide sufficient conditions such that this sys...

We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e. of the harmonic extension)...

This paper presents a multiscale method for the numerically efficient electromagnetic analysis of two-dimensional photonic and electromagnetic crystals. It is based on a contour integral method and a segmented analysis of more complex structures in terms of building blocks which are models for essential components. The scattering properties of esse...

Um in großen Lehrveranstaltungen effizient Übungsaufgaben mit Korrektur anbieten zu können bzw. Prüfungen automatisch korrigieren lassen zu können, bieten sich elektronische Systeme an. Wir stellen einige Vor- und Nachteile dar und beschreiben Möglichkeiten, wie man die automatische Bewertung qualitativ hochwertig gestalten kann.%%%%%%%%%%%%%%%%%%%...

We review approximation methods and their stability and applicability. We then focus on the finite section method and Galerkin methods and show that on separable Hilbert spaces either one can be interpreted as the other. In the end we demonstrate that well‐known methods such as the finite element method and polynomial chaos expansion are particular...

We describe a method to construct traces of quadratic forms in the
context of Hilbert spaces, relying on monotone convergence of forms and the
canonical decomposition into regular and singular part. This construction will
then be applied to the Bessel operator in one dimension.

We study the influence of certain geometric perturbations on the spectra of self-adjoint Schr\"odinger operators on compact metric graphs. Results are obtained for radially symmetric vertex conditions, which, amongst others, include $\delta$ and $\delta'$-type conditions. We show that adding edges to the graph or joining vertices changes the eigenv...

We consider operators $A$ on a sequentially complete Hausdorff locally convex space $X$ such that $-A$ generates a (sequentially) equicontinuous equibounded $C_0$-semigroup. For every Bernstein function $f$ we show that $-f(A)$ generates a semigroup which is of the same `kind' as the one generated by $-A$. As a special case we obtain that fractiona...

This article deals with a variation of constants type inequality for semigroups acting consistently on a scale of Banach spaces. This inequality can be characterized by a corresponding (easy to verify) inequality for their generators. The results have applications to heat kernel estimates and provide a unified perspective to estimates of these type...

We consider the linearised Korteweg–deVries equation, sometimes called Airy equation, on general metric graphs with edge lengths bounded away from zero. We show that properties of the induced dynamics can be obtained by studying boundary operators in the corresponding boundary space induced by the vertices of the graph. In particular, we characteri...

This contribution reports on implementing computer-assisted exams within a course on numerical analysis for engineering students. As in many courses on applied mathematics, in order to give a glimpse on realistic problems one is faced with large computations which are typically done by computers. However, when it comes to exams on such topics stude...

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $\ell^p (\ZM)$. Here, we gene...

We generalise the notion of band-dominated operators originally introduced for the space ℓp(ℤⁿ) to the setup of metric measure spaces and show various algebraic properties of this space of operators. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

We consider the linearized Korteweg-de-Vries equa- tions, sometimes called Airy equation, on general metric graphs with edge lengths bounded away from zero. We show that pro- perties of the induced dynamics can be obtained by studying boundary operators in the corresponding boundary space indu- ced by the vertices of the graph. In particular, we ch...

On an infinite, radial metric tree graph we consider the corresponding Laplacian equipped with self-adjoint vertex conditions from a large class including $\delta$- and weighted $\delta'$-couplings. Assuming the numbers of different edge lengths, branching numbers and different coupling conditions to be finite, we prove that the presence of absolut...

We elaborate a new method for constructing traces of quadratic forms in the framework of Hilbert and Dirichlet spaces. Our method relies on monotone convergence of quadratic forms and the canonical decomposition into regular and singular part. We give various situations where the trace can be described more explicitly and compute it for some illust...

We consider fractional powers of non-negative operators in Banach spaces defined by means of the Balakrishnan operator. Under mild assumptions on the operator we show that the fractional powers can also be obtained by a generalised Dirichlet-to-Neumann operator for a Bessel-type differential equation.

In order to increase the motivation and active learning of first-year students in the mathematics class throughout the course of the semester, links between mathematics and engineering science are demonstrated by means of mechanical engineering problems. These are designed such that they purposefully require knowledge and skills from mathematics an...

We consider Sturm-Liouville operators with measure-valued weight and potential, and positive, bounded diffusion coefficient which is bounded away from zero. By means of a local periodicity condition, which can be seen as a quantitative Gordon condition, we prove a bound on eigenvalues for the corresponding operator in Lp, for 1≤p<∞\documentclass[12...

We review recent developments in the spectral theory of continuum one-dimensional quasicystals, yielding purely singular continuous spectrum for these Schr\"odinger operators. Allowing measures as potentials we can generalize some results to very singular potentials, including Kronig-Penney type models.

In the present paper the Airy operator on star graphs is defined and studied. The Airy operator is a third order differential operator arising in different contexts, but our main concern is related to its role as the linear part of the Korteweg-de Vries equation, usually studied on a line or a half-line. The first problem treated and solved is its...

Ingenieurstudierende haben vor allem in der Studieneingangsphase oftmals Probleme mit den Mathematikveranstaltungen, die auch daher stammen, dass Mathematik für viele Studiengänge gemeinsam und damit notgedrungen mit nur wenig Anwendungsbezug unterrichtet wird. Wir beschreiben in diesem Artikel sowohl allgemein als auch auf unseren speziellen Fall...

At many universities, undergraduate courses of fundamental subjects such as mathematics are taught to students enrolled in many different course programs. Since the fundamental subject is then taught without using examples from the students’ main subjects, this often results in low student motivation in the fundamental subject and
lacking knowledge...

We show that for a large class of equivariant continuous, possibly non-selfadjoint, operators over minimal dynamical systems the spectrum as a set is in fact independent of the random parameter. Furthermore, the spectrum agrees with the essential spectrum. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

We give a characterization of a variation of constants type estimate relating two positive semigroups on (possibly different) (Formula presented.)-spaces to one another in terms of corresponding estimates for the respective generators and of estimates for the respective resolvents. The results have applications to kernel estimates for semigroups in...

We review recent developments in the spectral theory of continuum one-dimensional quasicystals, yielding purely singular continuous spectrum for these Schrödinger operators. Allowing measures as potentials we can generalize some results to very singular potentials, including Kronig-Penney type models.

We consider equivariant continuous families of discrete one-dimensional
operators over arbitrary dynamical systems. We introduce the concept of a
pseudo-ergodic element of a dynamical system. We then show that all operators
associated to pseudo-ergodic elements have the same spectrum and that this
spectrum agrees with their essential spectrum. As a...

We give a characterization of a certain estimate relating two positive
semigroups on general Banach lattices to one another in terms of corresponding
estimates for the respective generators and of estimates for the respective
resolvents. The resuts have applications to kernel estimates for semigroups
induced by accretive and non-local forms on $\si...

Given a positive C0-semigroup T0 on L2(Ω, m) with a kernel k0, where (Ω, m) is a σ-finite measure space, we study a suitably perturbed semigroup T and prove existence of a kernel k for T and an estimate of the k in terms of k0. In this way we extend a heat kernel estimate proven by Barlow, Grigor’yan and Kumagai [4] for Dirichlet forms perturbed by...

We prove a quantitative version of Gordon's Theorem concerning absence of
eigenvalues for Jacobi matrices and Sturm-Liouville operators with complex
coefficients.

We describe singular diffusion in bounded subsets $\Omega$ of $\mathbb{R}^n$
by form methods and characterize the associated operator. We also prove
positivity and contractivity of the corresponding semigroup. This results in a
description of a stochastic process moving according to classical diffusion in
one part of $\Omega$, where jumps are allow...

We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. As an application we consider quasiperiodic measures as potentials. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study Schr\"odinger operators on $\R$ with measures as potentials.
Choosing a suitable subset of measures we can work with a dynamical system
consisting of measures. We then relate properties of this dynamical system with
spectral properties of the associated operators.
The constant spectrum in the strictly ergodic case coincides with the union...

We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces
with respect to a (skew-)symmetric form, thus characterizing
(skew-)self-adjoint and unitary operators by means of self-ortho-gonal
subspaces. By orthogonality preserving mappings, these characterizations can be
transferred to abstract boundary value spaces of (skew-)symm...

We prove a perturbation result for positive semigroups, thereby
extending a heat kernel estimate by Barlow, Grigor'yan and Kumagai for
Dirichlet forms (\cite{bgk2009}) to positive semigroups. This also leads
to a generalization of domination for semigroups on $L_p$-spaces.

We study one-dimensional Schr\"odinger operators with complex measures as
potentials and present an improved criterion for absence of eigenvalues which
involves a weak local periodicity condition. The criterion leads to sharp
quantitative bounds on the eigenvalues. We apply our result to quasiperiodic
measures as potentials.

In this paper we prove that the existence of absolutely continuous spectrum
of the Kirchhoff Laplacian on a radial metric tree graph together with a finite
complexity of the geometry of the tree implies that the tree is in fact
eventually periodic. This complements the results by Breuer and Frank in
\cite{BreuerFrank2009} in the discrete case as we...

In this Dissertation thesis the spectral theory of Schrödinger operators modeling quasicrystals in dimension one ist investigated. We allow for a large class of measures as potentials covering also point interactions.
The main results can be stated as follows: If the potential can be very well approximated by periodic potentials, then the correspon...

We describe operators driving the time evolution of singular diffusion on finite graphs whose vertices are allowed to carry masses. The operators are defined by the method of quadratic forms on suitable Hilbert spaces. The model also covers quantum graphs and discrete Laplace operators.

The one-dimensional Schrödinger operator H=-Δ+V has eigenvalues if the potential V∈L 1,loc (ℝ) can be approximated by periodic potentials (in a suitable sense). The aim of this paper is to generalize this result to measures μ instead of potential functions V; i.e., to more singular potentials.

Given an arbitrary, finitely generated, amenable group, we consider ergodic Schrödinger operators on a metric Cayley graph with random potentials and random boundary conditions. We show that the normalised eigenvalue counting functions of finite volume parts converge uniformly. The integrated density of states (IDS) as the limit can be expressed by...