TU Dortmund
Fakultät für Mathematik


The use of H-distributions for obtaining a strong convergence
Jelena Aleksić
University of Novi Sad

We use different types of H-distributions to obtain possible strong convergence of a weakly convergent sequences in different spaces of functions (Lebesgue, Sobolev, Bessel...). This problem is of special interest when we investigate convergence of a sequence of approximate solutions to different classes of PDEs.

Kernels, Spectral Theory and the Dirichlet-to-Neumann Operator
Wolfgang Arendt
Institute of Applied Analysis, Ulm University

Many cases are known where a semigroup has a continuous kernel. We will present a new and very efficient characterization for such kernel to exist. This will be used to study positive semigroups on spaces of continuous functions. Examples are the Robin Laplacian but also the Dirichlet-to-Neumann operator. A particular challenge is irreducibility. Ultracontractivity plays a particular role, not only for the kernel but also for the asymptotic behavior of the eigenvalues. Weyl's formula holds for the Dirichlet-to-Neumann operator associated to the Laplacian and also the Laplacian perturbed by a potential. The talk is based on joint work with Tom ter Elst.
W. Arendt, T. ter Elst: The Dirichlet-to-Neumann operator on spaces of continuous functions. arXiv:1707.05556
W. Arendt, T. ter Elst: Ultracontractivity and Eigenvalues. Weyl's Law for the Dirichlet-to-Neumann Operator. Integral Equations and Operator Theory 88 (2017) 65-89

On the solvability of a system of forward-backward linear equations with unbounded operator coefficients
Nikita Artamonov
MGIMO-University, Moscow, Russia

Consider a system of forward-backward evolution linear equations \[ \begin{pmatrix} x'(t) \\ y'(t) \end{pmatrix}= \begin{pmatrix} A & -B \\ -C & -A^* \end{pmatrix} \begin{pmatrix} x(t) \\ y(t) \end{pmatrix},\quad \begin{aligned} x(0) &=x_0 \\ y(T) &=G x(T) \end{aligned}\quad t\in[0,T]. \] Here $A$ is an accretive (unbounded) operator, $B$, $C$, $G$ are self-adjoint (unbounded) non-negative operators. This kind of system arises, for example, in some optimal control problems
It is proved the solvability of the system and the solvability of the related differential operator Riccati equation in the collection of Banach spaces.
N.V. Artamonov, On the solvability of a system of forward-backward linear equations with unbounded operator coefficients. Mathematical Notes, 100 (5-6), 747-750 (2016).

Unimodular transformations of conservative L-systems
Sergey Belyi
Troy University (USA)

We study unimodular transformations of conservative $L$-systems. Classes $\mathfrak M^Q$, $\mathfrak M^Q_\kappa$, $\mathfrak M^{-1,Q}_\kappa$ that are impedance functions of the corresponding $L$-systems are introduced. A unique unimodular transformation of a given $L$-system with impedance function from the mentioned above classes is found such that the impedance function of a new $L$-system belongs to $\mathfrak M^{(-Q)}$, $\mathfrak M^{(-Q)}_\kappa$, $\mathfrak M^{-1,(-Q)}_\kappa$, respectively. As a result we get that considered classes (that are perturbations of the Donoghue classes of Herglotz-Nevanlinna functions with an arbitrary real constant $Q$) are invariant under the corresponding unimodular transformations of $L$-systems. We define a coupling of an $L$-system and a so called $F$-system and on its basis obtain a multiplication theorem for their transfer functions. In particular, it is shown that any unimodular transformation of a given $L$-system is equivalent to a coupling of this system and the corresponding controller, an $F$-system with a constant unimodular transfer function.
In addition, we derive an explicit form of a controller responsible for a corresponding unimodular transformation of an $L$-system. Examples that illustrate the developed approach are presented.
The talk is based on joint work with K. A. Makarov and E. Tsekanovskiĭ (see references below).
[1] S. Belyi, K. A. Makarov, E. Tsekanovskiĭ, A system coupling and Donoghue classes of Herglotz-Nevanlinna functions, Complex Analysis and Operator Theory, 10 (4), (2016), 835-880.
[2] S. Belyi, K. A. Makarov, E. Tsekanovskiĭ, On unimodular transformations of conservative L-systems, Operator Theory: Advances and Applications (to appear). ArXiv http://arxiv.org/abs/1608.08583

Discrete spectrum for Dirac bent and branching chains
Irina V. Blinova
ITMO University, St.Petersburg, Russia

We study Dirac operators on an infinite quantum graph of a bent chain form which consists of identical rings connected at the touching points by δ-couplings. It is established that the negativity of the coupling parameter is the necessary and sufficient condition for the existence of eigenvalues of the Dirac operator. The investigation is based on the transfer-matrix approach. It allows one to reduce the problem to an algebraic task. δ-couplings was introduced by the operator extensions theory method.

Inverse problems for boundary triples with applications
Malcolm Brown
Cardiff University

We discusses the inverse problem of how much information on an operator can be determined, or detected from ‘measurements on the boundary’. Our focus is on non-selfadjoint operators and their detectable subspaces (these determine the part of the operator ‘visible’ from ‘boundary measurements’).
We show results in an abstract setting, where we consider the relation between the M- function (the abstract Dirichlet to Neumann map or the transfer matrix in system theory) and the resolvent bordered by projections onto the detectable subspaces. More specifically, we investigate questions of unique determination, reconstruction, analytic continuation and jumps across the essential spectrum.
The abstract results are illustrated by examples of Schrodinger operators, matrix- differential operators and, mostly, by multiplication operators perturbed by integral oper- ators (the Friedrichs model), where we use a result of Widom to show that the detectable subspace can be characterized in terms of an eigenspace of a Hankel-like operator.

Dynamic Mode Decomposition for Non-autonomus Systems
Bojan Crnković
Department of mathematics, University of Rijeka

Numerical approximation of the Koopman operator gained attention in recent years because of the wide range of applications that was enabled by a data driven approach to approximation of spectral objects of Koopman operator. There are many different approaches to this problem but Dynamic Mode Decomposition (DMD) has shown to be a powerful tool for the analysis of nonlinear autonomous systems. However, existing DMD theory deals with autonomous systems and periodic and quasi-periodic forcing of the system. In this paper we propose an extension of Koopman operator framework from usual autonomous system setting to general non-autonomous setting with applications on linear and nonlinear dynamical systems. We propose a data driven algorithm based on DMD algorithm to obtain a time dependent finite dimensional approximation of Koopman operator.

A shear flow problem for compressible viscous and heat conducting micropolar fluid
Ivan Dražić
Faculty of Engineering, University of Rijeka

We consider the non-stationary shear flow between two parallel solid and thermoinsulated horizontal plates, with the upper one moving irrotationally. The fluid is compressible, micropolar, viscous and heat-conducting, as well as is in thermodynamical sense perfect and polytropic. We assume that, given a Cartesian coordinate system $x$, $y$ and $z$, solutions of corresponding problem are $x$-dependent only.
In this work we present the existence and uniqueness results for corresponding problem with non-homogeneous boundary data for velocity and homogeneous boundary data for microrotation and heat flux, under the additional assumption that the initial density and initial temperature are strictly positive.
This is the joint work with Loredana Simčić.

The inverse monodromy problem for canonical systems
Harry Dym
The Weizmann Institute of Science

In this talk I will present some highlights of joint investigations with D. Z. Arov over the past 20 plus years on direct and inverse problems for canonical systems of integral and differential systems and related applications. The talk will focus on the inverse monodromy problem for $m\times m$ canonical differential systems $$ y_t^\prime(\lambda)=i\lambda y_t(\lambda)H(t)J $$ on a finite interval $[0,d]$, where $H(t)$ is a summable $m\times m$ matrix valued function on $[0,d]$ that is positive semi-definite a.e. and $J$ is an $m\times m$ signature matrix.
The inverse problem is to recover the Hamiltonian $H(t)$ of the differential system from the monodromy matrix, i.e., from the value of the matrizant (fundamental solution) of the system at the right hand end point $d$ of the interval. This problem does not have a unique solution unless extra constraints are imposed. Some known results will be reviewed briefly. Special classes of monodromy matrices for which the solutions of the inverse monodromy problem may be parametrized by identifying the matrizant with the resolvent matrices of a class of bitangential extension problems will be discussed. The development makes extensive use of two classes of reproducing kernel Hilbert spaces of vector valued entire functions that originate in the work of Louis de Branges and the interplay between them. Some new subclasses of these spaces and their role in the inverse monodromy problem will also be discussed if time permits.

A coupling problem for entire functions
Jonathan Eckhardt
University of Vienna

I will discuss a coupling problem for entire functions which arises in inverse spectral theory for singular second order ordinary differential equations/two-dimensional first order systems and is also of relevance for the integration of certain nonlinear wave equations.

Null-controllability of the heat equation on rectangular regions
Michela Egidi
TU Dortmund

We consider the controlled heat equation on rectangular region $[0,2\pi L]^d$ with $d\geq 1$ and $L>0$ and control function acting on a subset $\omega\subset [0,2\pi L]^d$ of positive measure. We show that such a system is null-controllable at time $T>0$, i. e. there exists a control function such that the solution of the system is driven to zero at time $T$, and we give an estimate of the control cost in terms of the geometric parameter of the problem.

Friedrichs systems in a Hilbert space framework: solvability and multiplicity
Marko Erceg
Department of Mathematics, Faculty of Science, University of Zagreb

The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide sufficient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples. This is a joint work with Nenad Antonić and Alessandro Michelangeli.

Periodic quantum graphs with uncommon spectra
Pavel Exner
Czech Academy of Sciences

The usual expectation concerning the spectrum of a periodic quantum graph is a family of bands, some absolutely continuous, some degenerate, and gaps, typically infinitely many of them. The aim of this talk is to show that the picture could be different. This is illustrated using two simple examples, a chain of loops and a rectangular lattice. In particular, we are going to show that (i) a chain in a homogeneous magnetic field can have no absolutely continuous spectrum at all, (ii) a chain in a linear magnetic can have a spectrum of a fractal nature, and (iii) even without any external field a lattice can have a finite number of spectral gaps in analogy with Bethe-Sommerfeld behaviour of the `usual' Schrödinger operators. The last two effects depend on the number-theoretic properties of the model parameters.

On multi-qubit quantum channel quality estimation
Maria P. Faleeva
ITMO University, St.Petersburg, Russia

Multi-qubit quantum channel is considered. A possible error is related to the creation of entangled state during the transmission. A way of the channel quality estimation is suggested. It is based on the approximation of the unitary transfer-matrix of the channel by a Kronecker (tensor) product. A few examples are considered.

Differential operators and generalized trigonometric functions on fractal subsets of the real line
Uta Freiberg
uni stuttgart

Spectral asymptotics of second order differential operators of the form $d/d\mu d/dx$ on the real line are well known if $\mu$ is a self similar measure with compact support. We extend the results to some more general cases such as random fractal measures. Moreover, we give a representation of the eigenfunctions as generalized trigonometric functions. The results were obtained in collaboration with Peter Arzt.

On factorizations of differential operators and Hardy-Rellich-type inequalities
Fritz Gesztesy
Baylor University, Waco, Texas, USA

Abstract: We will illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, using this factorization method, we will derive a general inequality and demonstrate how particular choices of the parameters contained in this inequality yield well-known inequalities, such as the classical Hardy and Rellich inequalities as special cases. Actually, other special cases yield additional and apparently less well-known inequalities.
We will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators.
This is based on joint work with Lance Littlejohn and Michael Pang.

Asymptotic Methods in the Spectral Analysis of Singular Sturm-Liouville Operators on the Line
Daphne Gilbert
Dublin Institute of Technology

Using the method of subordinacy [1], the Weyl-Kodaira theorem [2], and multiplicity results to I.S. Kac [3], we construct a systematic method of spectral analysis for singular Sturm-Liouville operators on the line. The procedure enables the location, spectral type and multiplicity of the spectrum to be identified at all points on the line. If time allows we will also consider some simple examples and applications in order to illustrate the process.
[1] D.J. Gilbert, On subordinacy and spectral multiplicity for a class of singular differential operators, Proc. R. Soc. Edinburgh 128A, pp. 549-584, 1998.
[2] N. Dunford and J.T. Schwarz, Linear Operators, Part II, Chapter III.5, Interscience, 1963.
[3] I. S. Kac, On the multiplicity of the spectrum of a second order differential operator and the associated expansion, Isv. Akad. Nauk SSSR, Ser. Mat. 27 (1963), pp. 1081 - 1112 pp. (in Russian).

Oscillation theory for Jacobi operators with infinite fibers
Julian Großmann
Hamburg University of Technology

Oscillation theory is known for almost two centuries and was mainly introduced by Charles-François Sturm and Joseph Liouville. This Sturm-Liouville oscillation theory can be studied for continuous Hamiltonian systems and discrete Jacobi operators. Here we consider Jacobi operators with block entries in a von Neumann algebra with finite trace, namely in a II$_1$ setting, and develop a rotation calculus for the spectrum of this Jacobi operator.
This is joint work with Hermann Schulz-Baldes and Carlos Villegas-Blas.

Numerical Tensor Techniques for Multidimensional Convolution Products
Wolfgang Hackbusch
Max-Planck-Institut Mathematik in den Naturwissenschaften, Leipzig

The Numerical Tensor Calculus enables computations of high-dimensional objects, e.g., multivariate grid functions. In this lecture we start from an example in quantum chemistry involving six-dimensional kernel functions and explain the technique of Numerical Tensor Calculus with particular emphasis on the convolution operation. A second subject is the tensorisation technique which also applies to one-dimensional grid functions and allows to perform the convolution with a cost which may be much cheaper than the fast Fourier transform.

On Dirac operators with singular interactions
Markus Holzmann
Graz University of Technology

In this talk Dirac operators with singular interactions supported on surfaces are discussed. Such operators can be used in relativistic quantum mechanics as idealized models for Dirac operators with strongly localized potentials. After establishing self-adjointness of these operators, I will discuss the structure of their discrete and essential spectra and their convergence in the nonrelativistic limit.

The solution of the Gevrey smoothing conjecture for the non-cutoff homogenous Boltzmann equation for Maxwellian molecules.
Dirk Hundertmark
Karlsruhe Institute of Technology

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^3)\cap L\log L(\mathbb{R}^3)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular. So far, the best available results show that the solution becomes $H^{\infty}$ regular for positive times. Gevrey regularity is also known for weak solutions of the linearised Boltzmann equation, where one studies solutions close to a Maxwellian distribution, or under additional decay assumptions on the solutions. The main problem for establishing Gevrey regularity is that, in order to use the coercivity results on the non-cutoff Boltzmann collision kernel, one has to bound a non-linear and non-local commutator of the Boltzmann kernel with certain sub-exponential weights. We prove, under the sole assumption that the initial datum is in $L^1_2(\mathbb{R}^3)\cap L\log L(\mathbb{R}^3)$, i.e., finite mass, energy and entropy, that the weak solution of the homogenous Boltzmann becomes Gevrey regular for strictly positive times. The main ingredient in the proof is a new way of estimating the non-local and non-linear commutator.
Joint work with Jean-Marie Barbaroux, Tobias Ried, and Semjon Wugalter.

Collocation-quadrature methods for Cauchy singular integral equations with additional fixed singularities
Peter Junghanns
Chemnitz University of Technology

We present recent results on the stability of collocation-quadrature methods based on Chebyshev nodes and applied to a class of Cauchy singular integral equations with additional Mellin operators having fixed singularities. The necessity and sufficience of the stability conditions is proved by $C^*$-algebra techniques. For example, it turns out that the part of the approximating operator sequence, associated with the Mellin part of the original equation, is ``very close'' to the finite section of particular operators belonging to a $C^*$-algebra of Toeplitz operators. Finally, these stability results are checked by applying the mentioned methods to integral equations for the two-dimensional elasticity problems of a notched half-plane and of a crack at a circular cavity surface. In particular, we will see that numerical results published in the book on two-dimensional elasticity theory by A. I. Kalandiya for the second problem seem not to be correct.

Definitizable normal linear operators on Krein spaces
Michael Kaltenbäck
TU Vienna

We will deal with a generalization of the spectral theorem for normal operators on Hilbert spaces to normal, definitizable operators $N$ on Krein spaces $\mathcal K$. Definitizability means the existence of $p\in \mathbb C[x,y]$ such that \[ [p(\frac{N+N^*}{2},\frac{N-N^*}{2i})x,x] \geq 0 \] for all $x\in \mathcal K$. We consider the ideal $I$ generated by all such $p\in \mathbb C[x,y]$ and assume $I$ to be zero-dimensional, i.e.\ $\dim \mathbb C[x,y]/I <\infty$. Applying primary factorization from ring theory we have a unique decomposition $I=Q_1\cap\dots\cap Q_m$, where $Q_j$ is a primary ideal contained in a unique maximal ideal $Q_j\subseteq P_j := \{f\in \mathbb C[x,y] : f(a_j)=0\}$ for pairwise distinct $a_j \in \mathbb C^2$. We consider $V_{\mathbb R}(I) := \{a_j: a_j \in \mathbb R^2\}$ as a subset of $\mathbb C$ and $V(I)\setminus \mathbb R^2$ as a subset of $\mathbb C^2$. For $w\in V_{\mathbb R}(I)$ define the finite dimensional algebras $\mathcal A(w):=\mathbb C[x,y]/(P_j\cdot Q_j)$, where $j$ is such that $a_j=w$, and for $a\in V(I)\setminus \mathbb R^2$ define $\mathcal B(a):=\mathbb C[x,y]/Q_j$, where $j$ is such that $a_j=a$. Now we consider functions $\phi$ which are defined on $\sigma(N) \dot\cup (V(I)\setminus \mathbb R^2)$ such that $\phi(z) \in \mathcal A(z)$ for $z\in V_{\mathbb R}(I) \cap \sigma(N)$, $\phi(a) \in \mathcal B(a)$ for $a\in V(I) \setminus \mathbb R^2$, $\phi$ is complex valued for all other point, and such that for all non-isolated $w \in V_{\mathbb R}(I) \cap \sigma(N)$ the functions $\phi$ has a finite Taylor like approximation near $w$. Provided with pointwise addition and multiplication and some sort of conjugation these functions $\phi$ form a $*$-algebra $\mathcal F_N$, and we show the existence of a $*$-algebra homomorphism from $\mathcal F_N$ into $\{N,N^*\}''\subseteq B(\mathcal K)$.

Twisted waves, Lifshitz tails, and squared potentials
Werner Kirsch
FernUniversitaet Hagen

We consider a quantum waveguide (e.g. a long wire with small cross section. The waveguide is homogeneous, but twisted in a random way. We investigate the influence of the random twist an the spectral properties of the waveguide. We prove that the twist produces Lifshitz tails at the bottom of the spectrum, i.e. the spectral density decays exponentially rapidly near the ground state energy. Ina first step we reduce the problem to a one dimensional random Schrödinger operator whose potential is the square of of an alloy-type random potential. Consequently we also have to deal with squared random potentials.

Normal form for second order differential equations
Ilya Kossovskiy
University of Vienna/Masaryk University in Brno

We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a complete convergent normal form for this class of ODEs. The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE $y''=0$. For a generic ODE, we also provide a unique normal form. By doing so, we give a solution to a problem which remained unsolved since the work of Arnold. The method can be immediately applied to important classes of second order ODEs, in particular, the Painleve equations.

Spectral Estimates for Infinite Quantum Graphs
Aleksey Kostenko
University of Ljubljana and University of Vienna

We investigate the bottoms of the spectra and essential spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices.
The talk is based on a joint work with N. Nicolussi.

Vadim Kostrykin
Johannes Gutenberg-Universität Mainz


Waves and diffusion on metric graphs
Marjeta Kramar Fijavž
University of Ljubljana

We study the generation of cosine families by second order differential operators with general boundary conditions on $L^p(\mathbb{R}_+,\mathbb{C}^l)\times L^p([0,1],\mathbb{C}^m)$. The abstract results are used to show well-posedness of wave- and diffusion equations on networks/metric graphs. This is joint work with Klaus-Jochen Engel.

Spectral enclosures for non-self-adjoint extensions of symmetric operators
Matthias Langer
University of Strathclyde

Boundary triples or quasi boundary triples can be used to describe closed extensions of symmetric operators with the help of an abstract boundary operator. In this talk I shall discuss how the Weyl function corresponding to the quasi boundary triple and the abstract boundary operator can be used to obtain spectral enclosures for such closed extensions. In particular, I shall present sufficient conditions for extensions being m-sectorial or having spectrum contained in a parabolic-type region. The results are illustrated with elliptic operators with non-self-adjoint boundary conditions and with Schrödinger operators with $\delta$-potentials supported on hypersurfaces with complex-valued coefficients.
This talk is based on joint work with Jussi Behrndt, Vladimir Lotoreichik and Jonathan Rohleder.

Optimal control of parabolic equations by spectral decomposition
Martin Lazar
Univerity of Dubrovnik

This talk deals with an optimal control problems of Bolza type for a class of parabolic equations. The aim is to present a new methodology − based on a spectral decomposition of the operator governing the evolution of the system. Its implementation is described for a particular problem which consists in finding the initial datum that minimises a particular cost functional and ensures that the final state lies within a prescribed distance to a given target. The method leads us to an explicit expression of the optimal final state in terms of the given problem data. The obtained expression, combined with standard optimal control arguments enable construction of a one-shot algorithm providing an approximate solution. Its efficiency is assessed through numerical experiments. Application of the method to a distributed control problem will be discussed as well.

Finite Sections of the Fibonacci Hamiltonian
Marko Lindner
Hamburg University of Technology

The finite section method (FSM) aims to approximate the inverse of an infinite matrix $A$ by the inverses of growing finite submatrices along the main diagonal of $A$. We show that this works, with arbitrary cut-off points, for the Fibonacci Hamiltonian $A$ on the axis as well as on the half axis.
This is joint work with Hagen Söding (Hamburg).

Faber-Krahn inequalities for the Robin Laplacian on exterior domains
Vladimir Lotoreichik
Czech Academy of Sciences

We will discuss generalizations of the Faber-Krahn inequality for the $1^{\text{st}}$ eigenvalue of the Robin Laplacian with a negative boundary parameter on a complement $\Omega^{\rm ext} := \mathbb{R}^d\setminus\overline{\Omega}$ of a bounded set $\Omega\subset\mathbb{R}^d$. Here our focus will be on three types of constraints: $|\Omega| = {\rm const}$, $|\partial\Omega| = {\rm const}$, or fixed averaged mean curvatures of $\partial\Omega$. We will present results for $\Omega$ being:

  1. a bounded, convex domain in $\mathbb{R}^d$, $d \ge 3$;
  2. a bounded domain in $\mathbb{R}^2$ with $N\in\mathbb{N}$ simply connected components;
  3. an open arc in $\mathbb{R}^2$ with two endpoints;
and some counterexamples in other geometric settings. The analysis of the type (c) domains relies on the min-max and the Birman-Schwinger principles. For the domains of the types (a) and (b), it is efficient to express the min-max principle on the level of quadratic forms in suitable coordinates on $\Omega^{\rm ext}$. In particular, for the type (b) domains, we use the coordinates on $\Omega^{\rm ext}$ employed by Payne and Weinberger (1961) for the estimate of the $1^{\text{st}}$ Dirichlet eigenvalue on a bounded domain. Remarkably, the Payne-Weinberger trick for the Robin Laplacian with a negative boundary parameter is more efficient for exterior domains than for bounded domains.
These results are obtained largely in collaboration with David Krejčiřík.

To the spectral theory of infinite quantum graphs
Mark Malamud
Institute Applied Mathematics and Mechanics, NAS Ukraine

Infinite quantum graphs with $\delta$-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a quantum graph and a certain discrete Laplacian given on a graph with infinitely many vertices and edges is established. In particular, it is shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete, etc. only simultaneously.
Spectral properties (e.g., selfadjointness, semiboundedness from below, discreteness property, spectral types, etc.) of the quantum graph with $\delta$-interactions will be discussed.
The talk is based on joint results with with P. Exner, A. S. Kostenko, H. Neidhardt announced in [1].
[1] P. Exner, A. S. Kostenko, M. Malamud, and H. Neidhardt, Infinite Quantum Graphs, Doklady Mathematics, (2017), Vol. 95, No. 1, 31-36.

The essential numerical range for unbounded linear operators
Marco Marletta
Cardiff University

We introduce the essential numerical range $W_e(T)$ for closable Hilbert space operators $T$ and study its properties including possible equivalent characterizations and perturbation results. Although many of the properties known for the bounded case do not carry over to the unbounded case, the essential numerical range $W_e(T)$ allows us to describe the set of spectral pollution in a unified way when approximating $T$ by the Galerkin method or domain truncation method. If time permits we shall discuss some preliminary results for operator pencils.

Pseudospectral functions of symmetric systems with the maximal deficiency index
Vadim Mogilevskii
Poltava V.G. Korolenko National Pedagogical University, Poltava, Ukraine

We consider the symmetric differential system \[ J y'-A(t)y=\lambda \Delta (t) y \] with $n\times n$-matrix coefficients $J(=-J^*=-J^{-1})$ and $A(t)=A^*(t), \; \Delta (t)\geq 0$ defined on an interval $[a,b) $ with the regular endpoint $a$. Let $\varphi (\cdot,\lambda)$ be a matrix solution of this system of an arbitrary dimension and let \begin{equation*} (Vf)(s)=\widehat f(s):=\int\limits_\mathcal I \varphi^*(t,s)\Delta(t)f(t)\,dt \end{equation*} be the Fourier transform of the function $f(\cdot)\in L_\Delta^2(\mathcal I)$. A pseudospectral function of the system is defined as a matrix-valued distribution function $\sigma(\cdot)$ of the dimension $n_\sigma$ such that $V$ is a partial isometry from $L_\Delta^2(\mathcal I)$ to $L^2(\sigma;\mathbb C^{n_\sigma})$ with the minimally possible kernel.
It is assumed that the deficiency indices $N_\pm$ of the system satisfies $N_-\leq N_+ = n$. For this case we define the monodromy matrix $B(\lambda)$ as a singular boundary value of a fundamental matrix solution $Y(t,\lambda)$ at the endpoint $b$ and parameterize all pseudospectral functions $\sigma(\cdot)$ of any possible dimension $n_\sigma\leq n$ by means of the the linear-fractional transform \begin{equation*} m_\tau(\lambda)=(C_0(\lambda)w_{11}(\lambda)+C_1(\lambda)w_{21}(\lambda))^{-1} (C_0(\lambda)w_{12}(\lambda)+C_1(\lambda)w_{22}(\lambda)) \end{equation*} and the Stieltjes inversion formula for $m_\tau(\lambda)$. Here $w_{ij}(\lambda)$ are the matrix coefficients defined in terms of $B(\lambda)$ and $\tau=\{C_0(\lambda), C_1(\lambda)\}$ is a Nevanlinna matrix pair (a boundary parameter) satisfying certain admissibility conditions. It turns out that the matrix $W(\lambda)=(w_{ij}(\lambda))_{i,j=1}^2$ has the properties similar to those of the resolvent matrix in the extension theory of symmetric operators.
The obtained results develop the results by Arov and Dym; A. Sakhnovich, L. Sakhnovich and Roitberg; Langer and Textorius.
The results of the talk are partially specified in [1], [2]
1. V.I.Mogilevskii, Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation, Methods Funct. Anal. Topology 21 (2015), no. 1
2. V.I.Mogilevskii, Spectral and pseudospectral functions of Hamiltonian systems: development of the results by Arov-Dym and Sakhnovich, Methods Funct. Anal. Topology 21 (2015), no. 4, 370-402.

Solvability of the operator Riccati equation in the Feshbach case
Alexander K. Motovilov
Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia

We consider a block operator matrix of the form $$ L=\left(\begin{array}{cc} A_0 & B \\ B^* & A_1 \end{array} \right), $$ where $A_0$ and $A_1$ are self-adjoint operators in Hilbert spaces $\mathfrak{H}_0$ and $\mathfrak{H}_1$, respectively, and $B$ is a bounded operator from $\mathfrak{H}_1$ to $\mathfrak{H}_0$. It is assumed that the operator matrix $L$ is in the Feshbach case, that is, the spectrum of the entry $A_0$ overlaps the absolutely continuous spectrum of the entry $A_1$. In addition, it is supposed that $B$ is such that the Schur complement $M(z)=A_0-z+B(A_1-z)^{-1}B^*$ admits analytic continuation in $z$ to certain domains adjacent to the absolutely continuous spectrum of $A_1$ in the so-called unphysical sheets of the spectral parameter plane and, moreover, that the continued operator-function $M(z)$ admits factorization in the sense of Markus and Matsaev (see [1]). The corresponding operator roots of $M(z)$ allow us to construct solutions to the operator Riccati equations associated with the block operator matrices $L'$ emerging from $L$ as a result of the so-called complex rotation of the entry $A_1$. In particular, the complex scaling of $A_1$ is allowed if $A_1=-\Delta+V$ where $\Delta$ is the Lapalacian in $\mathbb{R}^n$ and $V$ an appropriate analytic potential. The graph representation of invariant subspaces of $L'$ associated with resonance (sub)sets of $L$ is discussed.

[1] R. Mennicken and A. K. Motovilov, Operator interpretation of resonances arising in spectral problems for ${2}\times{2}$ operator matrices, Math. Nachr. 201 (1999), 117-181.

Surgery of quantum graphs: eigenvalues and heat kernels
Delio Mugnolo
Institute of Analysis - University of Hagen

Quantum graphs are collections of intervals glued at their endpoints in a network-like fashion, along with differential operators acting upon them. It is natural to consider elliptic equations associated with these structures: while the eigenvalues of quantum graph Laplacians may in principle be found as the zeros of a (trascendental) secular equation, this task is hard to pursue even for very simple quantum graphs, like stars. Since a pioneering paper by Nicaise in 1987, much attention has been devoted to derive a-priori spectral estimates that only depend on global quantities of combinatorial (like edge connectivity, total edge or vertex number), metric (like total length or diameter) or hybrid (like girth or the Cheeger constant) radius. We well review some recent advances in this field, especially those based on simple surgery principle that allow for spectral comparison of two different quantum graphs. Some of these ideas will also be extended to investigate changes of heat kernels upon graph operations.
This is joint work with Gregory Berkolaiko, James Kennedy and Pavel Kurasov.

On an Elasto-Acoustic Transmission Problem.
Rainer Picard
TU Dresden, Dresden, Germany

We consider a coupled system, where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes. Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions. The results are obtained in a Hilbert space setting incurring no regularity constraints on the boundary and the interface of the underlying regions.

Weyl Asymptotic Formula For Infinite Order Pseudo-differential Operators
Stevan Pilipović
University of Novi Sad

Joint work with B. Prangoski (University of Skopje) and J. Vindas (University of Gent) }

I will present our joint results on the Weyl asymptotic formulae for the operators that are not of power-log-type as in the finite order (distributional) setting, but of log-type, which in turn yields that the eigenvalues of infinite order $\Psi$DOs, with appropriate assumptions, are "very sparse". As a by-product of our analysis, we also obtain Weyl asymptotic formulae for a class of finite order Shubin $\Psi$DOs with some conditions on the symbols that are not the ones usually discussed in the literature. Moreover, I will present infinite order Sobolev type spaces $H^*_{A_p,\rho}(f)$, where $A_p$ stend for Gevrey type sequences, while $\rho>0.$ $H^*_{A_p,\rho}(f)$ satisfies most of the familiar results for the classical, finite order, Sobolev spaces
Key words and phrases: Ultradistributions, infinite order pseudo-differential operators, infinite order Sobolev spaces
2010 Mathematics Subject Classification: 35S05, 46F05, 47D03

Quantum graph with semi-infinite edges: resonance states completeness
Anton I. Popov
ITMO University, St.Petersburg, Russia

Resonance states completeness problem for quantum graphs of various geometry having two semi-infinite edges is considered. Delta-coupling is assumed at the vertices. The problem is reduced to the scalar factorization problem in Sz.-Nagy functional model. Scattering problem is considered in the framework of Lax-Phillips approach.

Time dependent graph: Schrodinger, wave and Dirac dynamics
Igor Y. Popov
ITMO University, St.Petersburg, Russia

Star-like quantum graph with delta-coupling at the vertex is considered. It is assumed that the lengths of edges vary in time. We deal with three different cases: Schrodinger equation on edges, wave equation on edges and Dirac equation on edges. Wave dynamics for these cases are compared. It is also made for another graph – a loop with attached segment. Dependence of the dynamics of the model parameters (delta-coupling parameter, type of length variation in time) is investigated.

Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces
Andrea Posilicano
DiSAT - Sezione di Matematica, Università dell'Insubria, Como, Italy

We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $\Sigma\subseteq\Gamma =\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. Such a result follows from a uniqueness result in inverse scattering for Schrödinger operators with singular $\delta$-type potential supported on the surface $\Gamma$ and of strength $\alpha\in L^{p}(\Gamma)$, $p>2$.

Approximation of Laplacians on fractals and other metric spaces by discrete Laplacians
Olaf Post
Universität Trier

We show a norm convergence result for the Laplacian on a class of post-critical fractals approximated by a sequence of finite-dimensional graph Laplacians. Similar arguments can also be used to approximate other Laplacians on suitable metric spaces by Laplacians on discrete weighted graphs.
As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm (joint work with Jan Simmer, University of Trier).

Structured matrices and bivariate polynomials
Karla Rost
Chemnitz University of Technology

Inverses of structured matrices, namely Toeplitz, Hankel, and Toeplitz-plus-Hankel matrices, can be suitable introduced as polynomials of two variables, called Bezoutians. How to obtain the coefficients of the involved polynomials and matrix representations of the inverses is discussed. Moreover, recent results of joint papers with Torsten Ehrhardt concerning the reverse problem - the inversion of Bezoutians - are presented.

Solution to a stochastic pursuit model using moment equations
Miroslava Růžičková
University of Białystok, Faculty of Mathematics and Informatics, Białystok, Poland

This is a joint work with Irada Dzhalladova. In the contribution is investigated the navigation problem of following a moving target, using a mathematical model described by a system of differential equations with random parameters in the form \begin{align} \begin{split} \dot{x}_1(t)&=\quad\quad\qquad\qquad\,\,-x_2(t)+b(\xi(t)),\\ \dot{x}_2(t)&=\alpha x_1(t)-(\delta+\beta)x_2(t)+\beta b(\xi(t)), \end{split} \end{align} where $\xi(t)$ is a Markov process with two possible states.
The differential equations, which employ controls for following the target, are solved by a new approach using moment equations. The origin of the theory of moment equations and their use in the examination of the properties of solutions can be found in the works by K. Valeev and his scientific school, see, for instance, [5]. The moment equations method was used also in works [1] - [4] for studying stability of solutions to various kinds of systems with random structure.
The main result of the contribution provides us with a guarantee that a solution to derived system of ordinary linear differential equations - the moment equations, behaves as the mean value of a solution to the stochastic model of pursuit of a target. Therefore, instead of a solution to a complicated system of differential equations with random parameters we can solve the system of ordinary differential equations by known methods.
Simulations are presented to test effectiveness of the approach.
[1] J. Diblík, I. Dzhalladova, M. Michalková, M. Růžičková, Modeling of applied problems by stochastic systems and their analysis using the moment equations, Adv. Difference Equ., (2013), 2013:152, 12 pp.
[2] J. Diblík, I. Dzhalladova, M. Michalková, M. Růžičková, Moment equations in modeling a stable foreign currency exchange market in conditions of uncertainty, Abstr. Appl. Anal., Vol. 2013 (2013), Art. ID 172847, 11 pp.
[3] I. Dzhalladova, M. Růžičková, V. Štoudková Růžičková, Stability of the zero solution of nonlinear differential equations under the influence of white noise, Adv. Difference Equ., (2015), 2015:143, 11 pp.
[4] M. Růžičková, I. Dzhalladova, The optimization of solutions of the dynamic systems with random structure, Abstr. Appl. Anal., (2011), Art. ID 486714, 18 pp.
[5] K. G. Valeev, I. Dzhalladova, Optimization of Random Processes, KNEU, Kiev, 2006. (in Russian)

Three bounds for non-real eigenvalues of singular indefinite Sturm-Liouville operators
Philipp Schmitz
TU Ilmenau

In this talk Sturm-Liouville operators associated to singular indefinite differential expressions of the form \[ \mathrm{sgn}(x) \left(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+q(x)\right),\quad x\in\mathbb R, \] with a potential $q\in L^1(\mathbb R)$ are considered. Due to the weight $\mathrm{sgn}$, a self-adjoint realisation in the Hilbert space $L^2(\mathbb R)$ does not exist but it exist in a Krein space. Such an operator may have non-real spectrum. In the case $q\in L^1(\mathbb R)$ its essential spectrum is real. Hence, the non-real spectrum consists only of isolated eigenvalues. By adapting techniques of mathematical physics (Birman-Schwinger principle and WKB method) to the related eigenvalue problem we obtain three different bounds for the non-real point spectrum. These bounds depend only on the potential $q$. This talk is based on a joint work with Jussi Behrndt (TU Graz) and Carsten Trunk (TU Ilmenau).

Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over finitely generated amenable groups
Christoph Schumacher
TU Dortmund

In [1], a Glivenko-Cantelli theory for almost additive functions on lattices was developed, which allows to perform uniform thermodynamic limits of monotone quantities like eigenvalue counting functions. We present a vast generalization of the results to finitely generated amenable groups replacing lattices. The main tools are Glivenko-Cantelli theory and Ornstein-Weiss quasi-tilings, which substitute for tilings. In the proof, we need to join the worlds of probability and geometric group theory.
All results are joint work [2] with F. Schwarzenberger and I. Veselić.
[1] C. Schumacher, F. Schwarzenberger and I. Veselić. A Glivenko-Cantelli theorem for almost additive unctions on lattices, Stoch. Proc. Appl., 127 (1) 179-208, 2017
[2] C. Schumacher, F. Schwarzenberger and I. Veselić. Glivenko-Cantelli Theory, Ornstein-Weiss Quasi-Tilings, and Uniform Ergodic Theorems for Distribution-Valued Fields over Amenable Groups, preprint, submitted, 2017

Glivenko-Cantelli Theorem for Almost Additive Funtions on Lattices
Fabian Schwarzenberger
HTW Dresden

We present a Glivenko Cantelli Theorem for a class of monotone, almost additive functions defined on lattices. While classical Glivenko Cantelli results deal with independent identical distributed random variables, we allow dependencies, associated to the underlying geometry of the lattice. Our result not only show convergence, but also give quantitative estimates on the speed of convergence. One main application we have in mind is the convergence of the eigenvalue counting functions for the lattice. Here our result shows the uniform existence of the associated limit function, the integrated density of states. The presented results are published in a joint work with C. Schumacher and I. Veselić [1]. In a subsequent work [2] (also presented on the conference) we generalize this approach to amenable groups.
[1] C. Schumacher, F. Schwarzenberger and I. Veselić. A Glivenko-Cantelli theorem for almost additive functions on lattices, Stoch. Proc. Appl., 127 (1) 179-208, 2017
[2] C. Schumacher, F. Schwarzenberger and I. Veselić. Glivenko-Cantelli Theory, Ornstein-Weiss Quasi-Tilings, and Uniform Ergodic Theorems for Distribution-Valued Fields over Amenable Groups, preprint, submitted, 2017

On the subspace perturbation problem
Albrecht Seelmann
Technische Universität Dortmund

The variation of closed subspaces associated with isolated components of the spectra of linear self-adjoint operators under a bounded additive perturbation is considered. Of particular interest is the least restrictive condition on the norm of the perturbation that guarantees that the maximal angle between the corresponding subspaces is less than π/2. This problem has been discussed by different authors and is still unsolved in full generality. We give a survey on the most common approaches and recent developments

Essential spectra of operators on goups via limit operator techniques
Christian Seifert
TU Hamburg, Institut für Mathematik

We study the essential spectrum of so-called band-dominated operators on $L_p(G,\mu)$, where $G$ is a group, $\mu$ is a suitable measure on $G$ and $p\in (1,\infty)$. Such operators are norm-limits of band-operators, i.e. operators of finite range interaction. We characterize the Fredholm property by invertibility of the limit operators and obtain norm estimates of the essential norm of an operator on terms of the norm of its limit operators. The results generalize previous ones for operators on $\ell_p(\mathbb{Z}^N)$ to an abstract setting which also includes the study of Toeplitz operators in Bergman-like spaces on $\mathbb{C}^d$ w.r.t. a gaussian measure.
This is joint work with Raffael Hagger (Leibniz Universität Hannover, Germany).

On an eigenvalue estimates for Quantum Graphs
Andrea Serio
Stockholm University

On the metric graph $\Gamma$ consider the Schrödinger operator $L_q = -\Delta + q$ with some electric potential $q \in L_1(\Gamma)$. An upper estimate for the spectrum was obtained by Berkolaiko, Kennedy, Kurasov and Mugnolo in the case $q \equiv 0$. We show that this estimate is in fact sharp by providing an explicit family of graphs and give a clear explanation by studying the related eigenfunctions. In the second part we present a generalization of the inequality in the case $q \not\equiv 0$.
The talk is based on joint work with Pavel Kurasov, Stockholm University.

Comparative index and Sturmian separation theorems for linear Hamiltonian systems
Roman Simon Hilscher
Masaryk University (Brno, Czech Republic)

This is a joint work with Peter Šepitka. The comparative index was introduced by J. Elyseeva (2007) as an efficient tool in matrix analysis, which has fundamental applications in the discrete oscillation theory. In this talk we discuss the implementation of the comparative index into the theory of continuous time linear Hamiltonian systems (including Sturm-Liouville differential equations of even order). As a result we obtain new Sturmian separation theorems as well as new and optimal estimates for left and right proper focal points of conjoined bases of these systems. We derive our results for general possibly abnormal (or uncontrollable) linear Hamiltonian systems. The results turn out to be new even in the case of completely controllable systems.

On non-self-adjoint Toeplitz matrices with purely real spectrum
Frantisek Stampach
Stockholm University

First, we show a sufficient condition on the symbol of a (possibly non-self-adjoint) Toeplitz matrix guaranteeing the spectrum of its principal submatrix of an arbitrary size is purely real. In case of banded Toeplitz matrices, we provide a complete characterization of the class of matrices with the above property. Second, we describe how the eigenvalues of the principal submatrices distribute on the real line as the size goes to infinity. The presented results will be demonstrated by concrete examples.

On completeness and basis properties of root vectors for a certain class of operator pencils
Ludmila Suhocheva
Voronezh State University


Dedicated to the memory of Thomas Azizov.

In this talk we discuss a well-known result by Azizov and Iohvidov regarding the completeness and basis properties of the root vector system of a completely continuous operator in Pontryagin space [1]. We also cover the applications of this result to the studies of completeness and basis properties of the root vector system of some operator-valued functions [2].
Consider a class of quadratic operator pencils $L$ $$ L(\lambda)=\lambda^2 A+\lambda B+C, $$ where $A$, $B$, $C$ are continuous self-adjoint operators acting on a Hilbert space $H$. Here $A$ is a completely continuous operator, $B=B_1+B_2$, where $B_1$ is uniformly positive, $B_2$ is a compact operator, and the spectrum of the operator $-B_1^{-1}C$ has no more than countable set of accumulation points. Moreover, the set of regular points of the pencil $L$ is non-empty. The criterion of double completeness in $H$ of the system of root vectors of $L$ is obtained for this class of pencils in terms of non-degeneracy of root subspaces and their linearizers. It turns out that the system of root vectors for such pencils is twice complete and twice basisness in $H$ simultaneously. In the case of double completeness of the system of root vectors in $H$, there exists an almost $[\cdot,\cdot]$-orthonormal Riesz basis in $H$ that is constructed using Jordan chain vectors of the pencil $L$.

[1] T.Ya. Azizov, I.S. Iohvidov. Mathematical studies, 6, Proceedings of Academy of Science of Moldova, Kishinev, 1971.
[2] T.Ya. Azizov, I.S. Iohvidov. Indefinite Inner Product Spaces, Science, Moscow, 1996.

Spectral estimates and an Ambartsumian Theorem for quantum graphs.
Rune Suhr
Stockholm University

It has been shown that if the spectrum of the Laplacian with standard conditions on a metric graph $\Gamma$ coincides with the spectrum of the Neumann-Neumann Laplacian on a finite interval, then $\Gamma$ is in fact just the interval. This can be seen as a geometric version of Ambartsumian's classical Theorem. We will present a generalization of this result to Schrödinger operators with electric potential $q \in L_1(\Gamma)$, and present examples that show that this result can not be generalized to generic vertex conditions. This is joint work with J. Boman and P. Kurasov.

Focal points of conjoined bases of linear Hamiltonian systems
Peter Šepitka
Masaryk University Brno, Czech Republic

In this talk we are dealing with oscillation properties of conjoined bases of linear Hamiltonian differential systems by employing the concept of the comparative index. The comparative index was introduced by J. Elyseeva as an efficient tool in matrix analysis, which has fundamental applications in the discrete oscillation theory. Recently the author jointly with R. Šimon Hilscher and independently J. Elyseeva implemented the comparative index into the theory of continuous time linear Hamiltonian systems. We derived new and optimal estimates for the numbers of left and right proper focal points of conjoined bases of these systems on bounded intervals. In this talk we show that for a given nondegenerate compact interval there exist conjoined bases with any number of left proper focals points and any number of right proper focal points in the range between explicitly given minimal and maximal values. We also present a construction of such a conjoined basis in terms of initial conditions at a given point.

The (integrated) density of states on Archimedean lattice graphs
Matthias Taeufer
TU Dortmund

We study the Laplace operator on the 11 existing Archimedean lattices, i.e. graphs, based on vertex-transitive tilings of the plane by regular polygons. It plays a role in understanding properties of materials such as graphene (Hexagonal lattice) or certain jarosites (Kagome lattice). Archimedean lattices may or may not exhibit eigenfunctions of finite support… The occurrence of finitely supported eigenfunctions corresponds to jumps in the Integrated Density of States (IDS) and is related to exotic material behavior.
Using Floquet theory, we can rather explicitely calculate the IDS (as well as the Density of States) of all 13 Archimedean lattices and thus identify all occurring finitely supported eigenfunctions.
This is joint work with Norbert Peyerimhoff.

Tiling theorems and application to random Schrödinger operators
Martin Tautenhahn
Friedrich Schiller-Universität Jena & Technische Universität Chemnitz

We study random Schrödinger operators in $L^2 (\mathbb{R}^d)$ of the type \[ H_\omega = -\Delta + V_\omega \] where $\Delta$ denotes the Laplace operator and $V : \Omega \times \mathbb{R}^{d} \to \mathbb{R}$ is a stationary jointly measurable Gaussian field on a complete probability space $(\Omega , \mathcal{F} , \mathbb{P})$. If the covariance function decays exponentially (but may change its sign arbitrary) we prove a Wegner estimate for finite volume restrictions of $H_\omega$.
In order to treat such indefinite correlations, we prove the following tiling theorem. Let $f \in L^1 (\mathbb{R}^d)$ be exponentially decaying. Then there are $C_1 \not = 0$ and $I_0 \in \mathbb{N}_0^d$ such that \[ \forall x \in \mathbb{R}^d \colon \quad \int_{\mathbb{R}^d} \frac{y^{I_0}}{C_1} f (x-y) \mathrm{d}y = 1 . \] This is a joint work with Ivan Veselić.

The Dirichlet-to-Neumann operator on rough domains
Tom ter Elst
University of Auckland

We consider a bounded connected open set $\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on $L_2(\Gamma)$. We show that the asymptotic behaviour of $S_t$ as $t \to \infty$ is related to properties of the trace of functions in $H^1(\Omega)$ which $\Omega$ may or may not have. We also show that they are related to the essential spectrum of the Dirichlet-to-Neumann operator.
The talk is based on a joint work with W. Arendt (Ulm).

Jacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator
Gerald Teschl
University of Vienna

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 Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.
Based on joint work with Tom Koornwinder and Aleksey Kostenko.

Spectrum of ${\mathcal P}{\mathcal T}$ symmetric operators
Carsten Trunk
TU Ilmenau, Institut für Mathematik, Ilmenau, Germany

A prominent class of objects studied in ${\mathcal P}{\mathcal T}$ symmetric quantum mechanic consists of the ${\mathcal P}{\mathcal T}$ symmetric Hamiltonians \begin{equation*} (\tau y) (x) := -y^{\prime\prime}(x) + x^2(ix)^\epsilon y(x), \quad \epsilon \geq 2, \mbox{ and }x\in\Gamma, \end{equation*} where $\Gamma$ is a contour in $\mathbb C$ which is, in general, different from the real line and satisfies, according to the rules of ${\mathcal P}{\mathcal T}$ symmetry, some additional conditions.
To the differential expression $\tau$ one can associate a ${\mathcal P}{\mathcal T}$ symmetric operator which is simultaneously selfadjoint in the Krein space $(L_2(\mathbb R),[\cdot,\cdot])$, where $[\cdot,\cdot]$ is given by $[\cdot,\cdot] := (\mathcal P\, \cdot,\cdot)$ and $(\cdot,\cdot)$ stands for the usual $L_2$-product, $\mathcal P$ is the parity.
We will show that the spectrum of such an operator consists of isolated eigenvalues only which accumulate at $\infty$. Moreover, we discuss the location of the (point) spectrum of such operators and we will determine areas in the complex plane which are free of eigenvalues. Contrary to physical intuition, we will single out many cases where the real axis does only contain finitely many eigenvalues, i.e. there are infinitely many eigenvalues in the non-real plane.
The talk is based on a joint work with Florian Leben (TU Ilmenau).

A geometric Iwatsuka type effect in quantum layers
Matěj Tušek
Czech Technical University in Prague (CTU Prague)

We consider magnetic Dirichlet Laplacian on a layer of a fixed width constructed along a two-dimensional hypersurface. The magnetic field will always be uniform, but the geometry of the layer may be non-trivial. If the layer is planar and the field is perpendicular to it the spectrum consists of infinitely degenerate eigenvalues. We will study translationally invariant geometric perturbations and derive several sufficient conditions under which the spectrum, in its entirety or a part of it, becomes absolutely continuous. The talk is based on a joint work with T. Kalvoda and P. Exner.

Space-time estimates for strongly propagative systems: From Maxwell to Dirac operators
Tomio Umeda
University of Hyogo

Spectral theory for general classes of first order systems has been less popular since 1990's. In this talk, I would like to propose an approach which can deal with Maxwell and Dirac operators in a unified manner, and introduce a new topic for these operators. Namely, I will discuss space-time estimates for these operators.
This talk is based on joint work with Matania Ben-Artzi (Hebrew University).

The Talbot effect and the dynamics of vortex filaments: transfer of energy and momentum
Luis Vega
Basque Center for Applied Mathematics, Bilbao

I shall present some recent work done in collaboration with V. Banica and F. De La Hoz about the evolution of vortex filaments according to so called binormal flow. I will exhibit a non-linear Talbot effect and some results concerning the transfer of energy and of linear momentum. The relation with Frisch-Parisi conjecture in turbulence will be also discussed.

Spectral analysis of functional differential equations in Hilbert space and its applications
Victor Vlasov and Nadezhda Rautian
Lomonosov Moscow State University

We study functional differential and integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the considered equations is an abstract hyperbolic equation perturbed by terms with delay and terms containing Volterra integral operators. We prove that the initial boundary-value problems in weighted Sobolev spaces are well posed on the positive semi-axis for the specified equations. We also research the spectra of operator-valued functions that are symbols of these equations in the autonomous case. We study the spectral problems for operator-valued functions that are symbols of Volterra integrodifferential equations with unbounded operator coefficients in Hilbert spaces. Operator models of such type have many applications in the linear viscoelasticity theory, homogenization theory, heat conduction theory in media with memory, etc.
[1] N.A. Rautian and V.V. Vlasov, Properties of solutions of integro-differential equations arising in heat and mass transfer theory, Transactions of the American Mathematical Society. 2014, vol. 75, рр. 185-204.

Heat kernels of non-symmetric jump processes: beyond the stable case
Zoran Vondraček
University of Zagreb

Let $J$ be the Lévy density of a symmetric Lévy process in $\mathbb{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $$ \mathcal{L}^{\kappa}f(x):= \lim_{\varepsilon \downarrow 0} \int_{\{z \in \mathbb{R}^d: |z|>\varepsilon\}} (f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$ where $\kappa(x,z)$ is a Borel function on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and$|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1]$. Following the work of Z.-Q. Chen and X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Relat. Fields 165, 267–312 (2016), who considered the case $J(z)=|z|^{-d-\alpha}$, $\alpha\in (0,2)$, we construct the heat kernel $p^\kappa(t, x, y)$ of $\mathcal{L}^\kappa$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^\kappa$. Our approach uses a combination of analytic and probabilistic methods.
Joint work with P. Kim (SNU) and R. Song (UIUC)

A functional model for maximally dissipative operators
Ian Wood
University of Kent

We introduce an abstract framework for a maximally dissipative operator A and its anti-dissipative adjoint and use this framework to construct the selfadjoint dilation of A using the Straus characteristic function. The advantage of this construction is that the parameters arising in the dilation are explicitly given in terms of parameters of A (such as coefficients of a differential expression). Having constructed the selfadjoint dilation, we will discuss its spectral representation. The abstract theory will be illustrated by the example of dissipative Schrodinger operators.

Directing functionals and de Branges space completions in almost Pontryagin spaces
Harald Woracek
Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria

The following theorem holds: Let $\mathcal L$ be a - not necessarily nondegenerated or complete - positive semidefinite inner product space carrying an anti-isometric involution, and let $S$ be a symmetric operator in $\mathcal L$. If $S$ possesses a universal directing functional $\Phi:\mathcal L\times\mathbb C\to\mathbb C$ which is real w.r.t. the given involution, and the closure of $S$ in the completion of $\mathcal L$ has defect index $(1,1)$, then there exists a de Branges (Hilbert-) space $\mathcal B$ such that $x\mapsto\Phi(x,\cdot)$ maps $\mathcal L$ isometrically onto a dense subspace of $\mathcal B$ and the multiplication operator in $\mathcal B$ is the closure of the image of $S$ under this map.
In this paper we consider a version of universal directing functionals defined on an open set $\Omega\subseteq\mathbb C$ instead of the whole plane, and inner product spaces $\mathcal L$ having finite negative index. We seek for representations of $S$ in a class of reproducing kernel almost Pontryagin spaces of functions on $\Omega$ having de Branges-type properties. Our main result is a version of the above stated theorem, which gives conditions making sure that $\Phi$ establishes such a representation. This result is accompanied by a converse statement and some supplements.
As a corollary, we obtain that if a de Branges-type inner product space of analytic functions on $\Omega$ has a reproducing kernel almost Pontryagin space completion, then this completion is a de Branges-type almost Pontryagin space. This is an important fact in applications. The corresponding result in the case that $\Omega=\mathbb C$ and $\mathcal L$ is positive semidefinite is well-known, often used, and goes back (at least) to work of M.Riesz in the 1920's.

Rank 2 Perturbations of Matrices and Spectral Properties
Janusz Wysoczanski
Institute of Mathematics, Wroclaw University

We present a general theory of rank 2 perturbations of matrices. Several special cases will be discussed. We shall also present the perturbatin of spectra. In particular the interlacing property and the limit behaviours will be discussed. The talk is based on the paper:
A. Kula. M. Wojtylak, J. Wysoczanski, "Rank 2 perturbation of matrices and operators and operator model for t-transformation of probability measures", J. Funct.Anal. 272 (2017), no.3

Rank 2 deformation of operators and their noncommutative distributions.
Anna Wysoczanska-Kula
Institute of Mathematics, Wroclaw University

We shall discuss the application of the theory of operators in problems related to noncommutative probability. In particular we shall discuss two special perturbations - the "diagonal" and the "antiiagonal" - which are related to some deformations of probability measures in free probability. The relation between operators and probability measures - their distributions - will be presented.
The talk is based on the paper: A. Kula. M. Wojtylak, J. Wysoczanski, "Rank 2 perturbation of matrices and operators and operator model for t-transformation of probability measures", J. Funct.Anal. 272 (2017), no.3

On square integrable solutions and principal solutions for linear Hamiltonian systems
Petr Zemánek
Masaryk University

Square integrable solutions and principal solutions play (separately) important roles in the qualitative theory of the linear Hamiltonian differential systems \begin{equation*}\tag{$\text{H}_\lambda$}\label{Hla} z'(t,\lambda)=[\mathcal{H}(t)+\lambda\,\mathcal{J}\,\mathcal{W}(t)]\,z(t,\lambda), \quad \mathcal{J}:=\begin{pmatrix} 0 & I\\ -I & 0\end{pmatrix}, \end{equation*} where $t\in[a,\infty)$, $\lambda\in\mathbb{C}$ is a spectral parameter, and $\mathcal{H}(t)$ and $\mathcal{W}(t)$ are piecewise continuous even order matrix-valued functions such that $\mathcal{J}\mathcal{H}(t)+\mathcal{H}^*(t)\mathcal{J}=0$ and $\mathcal{W}(t)=\mathcal{W}^*(t)\geq0$ for all $t\in[a,\infty)$. We present our recent results in the Weyl-Titchmarsh theory for system \eqref{Hla} which were derived by using principal solutions. In particular, we show a close connection between the Weyl solution and the principal solution; compare with Theorems 2.13 and 3.11 in [1] for the second order Sturm-Liouville differential equations. The talk is based on the joint research with R. Šimon Hilscher, see [2].
[1] S. L. Clark, F. Gesztesy, and R. Nichols, Principal solutions revisited, in "Stochastic and Infinite Dimensional Analysis", C. C. Bernido, M. V. Carpio-Bernido, M. Grothaus, T. Kuna, M. J. Oliveira, and J. L. da Silva (editors), Trends in Mathematics, pp. 85-117, Birkhäuser, Basel, 2016.
[2] R. Šimon Hilscher and P. Zemánek, On square integrable solutions and principal and antiprincipal solutions for linear Hamiltonian systems, submitted.

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