Operator matrices and delay equations

András Bátkai

Eötvös Loránd University Budapest, Hungary

We survey some connections between operator matrices, pencils and abstract delay differential equations.

On sectorial realizations of Stieltjes and inverse Stieltjes functions

Sergey Belyi

Troy University, USA

We study sectorial classes of Stieltjes and inverse Stieltjes functions acting on finite-dimensional Hilbert space as well as their scalar versions. It is shown that a function from these classes can be realized as the impedance function of an \(L\)–system whose either state–space or associated operator is sectorial. Moreover, it is established that the knowledge of the limit values of these functions allows us to find angles of sectoriality of the corresponding operators related to a realizing \(L\)–system. The exact angle of sectoriality of the accretive main operator \(T\) of such a system is also found and the corresponding formulas are provided. These results are illustrated by examples of the realizing \(L\)–systems based upon the Schrödinger operator on the half–line.

The talk is based on a recent joint work with Yu. Arlinskii and E. Tsekanovskii.

Characterization of generalized poles of generalized Nevanlinna functions by pole cancelation functions

Muhamed Borogovac

Boston Mutual Life, USA

In this paper, a new more general definition of pole cancelation functions \(\eta \left( z \right)\) at a generalized pole \(\alpha \in \mathbb{R}\) of a generalized Nevanlinna function \[Q\left( z \right)=Q\left( \bar{z}_{0} \right)+\left( z-\bar{z}_{0} \right)\Gamma^{+}\left( I-(z-z_{0} \right){(A-z)}^{-1})\Gamma\] is introduced and it is proven that for each such pole cancelation function of a given order \(l\) there exists a corresponding Jordan chain of the representing linear relation \(A\) of length \(\ge l\).

The converse statement is proven under the condition that the general pole \(\alpha \in \mathbb{R}\) is not a zero of \(Q\) in the same time. For a given Jordan chain \(x_{0},\thinspace x_{1},\thinspace \ldots x_{l-1}\)of \(A\) at eigenvalue \(\alpha \in \mathbb{R}\) it is proven that the function \[\eta \left( z \right):=\left( z-\bar{z}_{0} \right){Q\left( z \right)}^{-1}\Gamma^{+}\left\{ x_{0}+\left( z-\alpha \right)x_{1}+\ldots +{(z-\alpha )}^{l-1}x_{l-1} \right\}\] satisfies all requirements of the previously introduced definition of pole cancelation functions of order \(\ge l\).

If pole \(\alpha \in \mathbb{R}\) is also a zero of \(Q\), then pole cancelation function \(\eta \) need not to exist. An example is provided that shows this.

Moreover, it is shown how the methodology of characterization of generalized poles by our pole cancelation functions can be applied even in the case when \(\alpha \in \mathbb{R}\) is both pole and zero of \(Q\).

This is a joint work with Annemarie Luger.

Large coupling convergence

Johannes Brasche

Institute of Mathematics, TU Clausthal,
Clausthal-Zellerfeld, Germany

We discuss strong, uniform and trace norm convergence of the resolvents of the selfadjoint operators associated with closed forms of the form \(E+bP\), as the real parameter \(b\) tends to infinity. In particular, we present criteria for operator norm convergence and trace norm convergence with maximal rate, respectively.

Sturm-Liouville operator on the line with retarded potential

E.V. Cheremnikh

Lviv Polytechnic National University, Ukraine

We consider the operator where \(q(x)y(x)\) is replaced by \(q(x)y(x-a)\). Function \(q(x)\) is complex valued exponentialy decrising. Friedrichs’ model is applicable and some condition of finitness of point spectrum is obtain. (work with Diaba F. and Zemmouri A.)

A coupling problem for entire functions

Jonathan Eckhardt

University of Vienna, Austria

It is well known that the inverse scattering problem for a Schroedinger operator on the line can be reformulated as Riemann-Hilbert problems. This connection has important applications for the derivation of long-time asymptotics for integrable wave equations. However, if the underlying Lax operator has purely discrete spectrum, then this approach does not apply anymore. In this case, the role of the Riemann-Hilbert problem is taken by a certain coupling problem for entire functions.

Spectral properties and approximations of a class of rational operator functions

Christian Engström

Department of Mathematics and Mathematical Statistics,
Umeå University, Sweden

In this talk we present spectral properties and Galerkin approximations of a class of self-adjoint rational operator functions. The natural linearization of the rational operator function results in a block operator matrix formulation with non-separated diagonal entries. We determine the essential spectrum of this self-adjoint block operator matrix and establish variational principles. Moreover, we apply the new theory to an unbounded operator function with periodic coefficients. The main application for this operator function is metallic photonic crystals. These nano-sized structures can be used to control the flow of light and are for example used in integrated optics.

This is a joint work with Heinz Langer and Christiane Tretter.

Weak solutions for the Dirichlet problem associated to a general class of degenerate elliptic PDE

Aurelian Gheondea

Bilkent University, Ankara, Turkey and IMAR, Bucharest, Romania

We characterize the existence of weak solutions for the Dirichlet problem associated to a general class of degenerate elliptic PDE by means of triplets of closely embedded Hilbert spaces. (Joint work with P. Cojuhari.)

Eigenfunction Expansions Associated with the One-Dimensional Schroedinger Operator on the Line

Daphne Jane Gilbert

Dublin Institute of Technology, Ireland

We consider the one–dimensional Schroedinger operator \(H\) in the case where Weyl’s limit point case holds at both of the infinite end points. Use of the method of subordinacy, the Weyl-Kodaira form of the eigenfunction expansion and a theorem of Kac enables us to determine a \(2\times 2\) spectral density matrix, from which the multiplicity of the spectrum of \(H\) and explicit formulae for the eigenfunctions may be determined. In addition a simplified form of the eigenfunction expansion, which reflects the location and structure of the spectrum, will be obtained. Examples to illustrate the process will be given in cases where the spectrum has multiplicity \(1\) and/or \(2\), and the spectrum is absolutely continuous and/or singular.

Kato’s square root theorem in the numerical analysis of finite element method.

Luka Grubišić

Department of mathematics, Faculty of Science,
University of Zagreb, Croatia

Kato’s square root theorem states that a maximal accretive square root of a second order convection-diffusion-reaction type operator (assuming Dirichlet boundary condition on \(\Omega\subset \mathbb{R}^n\)) with bounded and measurable coefficients has the first order Sobolev space \(H^1_0(\Omega)\) as its domain of definition.

This problem has been open for 30 years and has been solved by Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philippe Tchamitchian in 2001.

We present a relative perturbation theory for the approximation of eigenvalues of such operators from a subspace. We apply our abstract results to obtain error estimators for finite element approximations of eigenvalues and eigenfunctions.

We present several benchmark examples to illustrate the theory.

This is a joint work with Stefano Giani, Agnieszka Miedlar and Jeff Oval.

Recent advances in oscillation theory of discrete symplectic systems

Roman Šimon Hilscher

Masaryk University, Czech Republik

Symplectic systems represent a discrete time analogue of linear Hamiltonian systems. They contain as special cases many important difference equations and systems, namely the Sturm–Liouville difference equations, symmetric three-term recurrence equations, Jacobi difference equations, and linear Hamiltonian difference systems. Following our recent work in Linear Algebra Appl. and SIAM J. Matrix Anal. Appl., we present a new theory of discrete symplectic systems, in which the dependence on the spectral parameter is nonlinear. This requires to develop new definitions of (finite) eigenvalues and (finite) eigenfunctions and their multiplicities for such systems. Our main results include the corresponding oscillation theorems, which relate the number of (finite) eigenvalues with the number of focal points of the principal solution in the given discrete interval, and comparison theorems for (finite) eigenvalues of two symplectic eigenvalue problems. The present theory generalizes several known results, which depend linearly on the spectral parameter. Our results are new even for the above mentioned special discrete symplectic systems.

Frequency concentration of the eigenfunctions of the Anderson model

Francisco Hoecker-Escuti

Technische Universität Chemnitz, Germany

In an important paper in 2002, W. Schlag, C. Shubin and T. Wolff proved that, in a small coupling constant regime \(\lambda \ll 1\), the mass of the eigenfunctions of the Anderson model \(H_\omega=\triangle + \lambda V_\omega\) concentrate essentially in rings of size \(\lambda^2\) in Fourier space. Their result is valid in dimensions 1 and 2 (in dimension 2 or larger one restricts the operator to a box). The purpose of this talk is to show that the same phenomenon appears in any dimension.

Inverse scattering for singular Schroedinger operators

Rostyslav Hryniv

Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine and
University of Rzeszów, Rzeszów, Poland

We discuss the question, for what classes of singular 1-dimensional Schroedinger operators the direct and inverse scattering problems have complete solutions. In particular, we describe the set of potentials for which the corresponding transformation operators are of Hilbert–Schmidt class.

Based on a joint project with Ya. Mykytyuk (Lviv).

Left invertibility of formal Hamiltonian operators

Junjie Huang

School of Mathematical Sciences, Inner Mongolia University,
Hohhot, PRC

In this talk, we investigate the left invertible completion of the partial formal Hamiltonian operators \(\left(\begin{smallmatrix} A & ? \\ 0 & -A^{\ast} \end{smallmatrix} \right)\) and \(\left(\begin{smallmatrix} A & ? \\ C & -A^{\ast} \end{smallmatrix} \right)\) with unbounded entries. In particular, the left invertible completion of the partial Hamiltonian operators are also given. Based on the space decomposition technique, the alternative sufficient and necessary conditions are given according to whether the dimension of \(\mathcal{R}(A)^{\bot}\) is finite or infinite.

This work were completed jointly with Alatancang and Yaru Qi.

Non-Autonomous Port-Hamiltonian systems

Birgit Jacob

Fachbereich C - Mathematik und Naturwissenschaften,
Arbeitsgruppe Funktionalanalysis,
Bergische Universität Wuppertal, Germany

In this talk we discuss the well–posedness of infinite–dimensional non-Autonomous port-Hamiltonian systems. Examples of dynamical systems in this class are wave and beam equations. The norm of such a system is given by the energy (Hamiltonian) of the system. This fact enables us to develop easy checkable necessary and sufficient conditions for the existence of solutions.

The Separation of two Matrices and its Application in the Perturbation Theory of Eigenvalues and Invariant Subspaces

Michael Karow

Institut für Mathematik, Technische Universität Berlin,
Berlin, Germany

We discuss the three definitions of the separation of two matrices given by Stewart, Varah and Demmel. Then we use the separation in order to obtain an inclusion theorem for pseudospectra of block triangular matrices. In the second part of the talk (which is joint work with Daniel Kressner), we present two perturbation bounds for invariant subspaces and compare them with the classical bounds of Stewart and Demmel.

The HELP inequality and the similarity problem for indefinite Sturm-Liouville operators

Aleksey Kostenko

University of Vienna, Austria

We study two problems. The first one is the similarity problem for indefinite Sturm–Liouville operators. The second object is the so-called HELP inequality, a version of the classical Hardy–Littlewood inequality proposed by W.N. Everitt in 1971.

Both problems are well understood when the corresponding Sturm-Liouville differential expression is regular. Our main main objective is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl–Titchmarsh \(m\)–functions at zero and at infinity. As a biproduct of this result we show that both problems are closely connected.

The \(\mathop{\mathrm{div}} A \mathop{\mathrm{grad}}\) without ellipticity

Vadim Kostrykin

Institut für Mathematik
Johannes Gutenberg-Universität Mainz, Germany

The talk discusses some recent results on \(\mathop{\mathrm{div}} A \mathop{\mathrm{grad}}\)–operator for sign–indefinite coefficient matrices \(A\). A simplest example of such kind is \(\mathcal{L}=-\frac{d}{dx}\mathop{\mathrm{sign}}(x)\frac{d}{dx}\) on a bounded interval. Using the representation theorem for indefinite quadratic forms, for a wide class of coefficient matrices we prove the existence of a unique self–adjoint, boundedly invertible operator \(\mathcal{L}\), associated with the form \(\langle \mathop{\mathrm{grad}} u, A \mathop{\mathrm{grad}} u\rangle\).

Low-rank tensor methods for high-dimensional eigenvalue problems

Daniel Kressner

MATHICSE, EPF Lausanne, Switzerland

We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector in a low-rank tensor format. In this talk, we provide an introduction and survey recent developments in this direction. Based on joint work with Christine Tobler.

Schrödinger operators with delta-potentials on manifolds

Christian Kühn

TU Graz, Austria

We will present an approach for the investigation of Schrödinger operators with delta-potentials supported on manifolds of codimension 2 or 3. The talk is based on a joint work with J. Behrndt, V. Lotoreichik and J. Rohleder.

Eigenspaces and spectra of nonnegative matrices under rank one perturbations

Leslie Leben

TU Ilmenau, Germany

In this talk, rank one perturbations of matrices in the space \((\mathbb{C}^n, [\cdot,\cdot])\) are studied. Here, \([\cdot,\cdot]\) is an indefinite inner product with an invertible symmetric matrix \(G\) as its Gramian, i.e. \([x,y] = \langle Gx, y \rangle\). Let \(A\) be a nonnegative matrix, i.e. \([Ax,x] \geq 0\), and let \(B\) be a symmetric matrix with respect to \([\cdot,\cdot]\), such that \(B-A\) is of rank one. We will give a full spectral description of \(B\) (depending on the spectrum of \(A\)), including all possible structures of the algebraic eigenspaces of \(B\). In particular, we show the estimate

\[\begin{aligned} \vert \dim \mathfrak{L}_0(A) - \dim \mathfrak{L}_0(B) \vert \leq 2,\end{aligned}\]

where \(\mathfrak{L}_0(A)\) is the algebraic eigenspaces of \(A\) at \(0\). As an illustration, the case of \((3\times3)\)-matrices will be discussed and all possible \(B\) will be described.

The talk is based on a joint work with F. Martinez Peria and C. Trunk.

Self-adjoint Laplacians on partitions with \(\delta\)- and \(\delta'\)-couplings

Vladimir Lotoreichik

Graz University of Technology, Austria

In the talk we will discuss self-adjoint Laplace operators acting on partitions of Euclidean spaces into finite number of bounded and unbounded Lipschitz domains. The problem becomes non-trivial when we pose boundary conditions which connect neighbouring domains in the partition. Spectral properties of the corresponding self-adjoint Laplacians turn out to be related to combinatorial properties of the partition such as the number of colours sufficient to colour all the domains in a way that any two neighbouring domains are associated with distinct colors. This talk is based on the joint work with Jussi Behrndt and Pavel Exner.

Perturbation determinants and trace formulas for singular perturbations

Mark Malamud

Institute of Applied Mathematics and Mechanics, NASU,
Donetsk, Ukraine

A new concept of a perturbation determinant for a pair of singularly perturbed operators will be introduced. Explicit formulas expressing the perturbation determinant in terms of boundary operators and the respective Weyl functions will be discussed.

The talk is based on a joint work with H. Neidhardt.

Inverse Scattering on the Half–Line for Energy–Dependent Schrödinger Equations

Stepan Manko

Current: Czech Technical University in Prague, Czech Republic
On leave from Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine

The talk is based on a joint project with R. Hryniv (Lviv, Ukraine). We develop the direct and inverse scattering theory for one-dimensional energy-dependent Schrödinger equations \[-y''+q(x)y+2kp(x)y=k^2y\] on the half-line with highly singular potentials \(q\), namely, we consider potentials \(q\) of the form \(q=u'+u^2\) for some \(u\in L^2(\mathbb{R}^+)\). Such potentials are called Miura potentials. Under some additional assumptions this Riccati representation of \(q\) is unique, and we study scattering problem for the above equation along with the energy-dependent boundary condition \[\cos\alpha y^{[1]}(0,k)+k\sin\alpha y(0,k)=0,\] where the quasiderivative \(y^{[1]}\) is defined as \(y^{[1]}(x,k):=y'(x,k)-u(x)y(x,k)\). We show that the mapping that to every problem determined by \((u,p,\alpha)\) puts into correspondence its scattering function \(S\) is continuous with continuous inverse. We also obtain an explicit reconstruction formula for \((u,p,\alpha)\) in terms of \(S\). To do so, we exploit the connection between the ZS-AKNS system and scattering with Miura potentials.

Normal projections in Krein spaces

Francisco Martínez Pería

Instituto Argentino de Matemática, Argentina

The aim of this talk is to present some results on (bounded), normal projections acting on Krein spaces, that is those projections acting on a Krein space that commute with their adjoints with respect to the indefinite inner product.

Although every normal projection acting on a Hilbert space turns out to be selfadjoint, this is not the case on Krein spaces. Indeed, a (closed) subspace is the range of a normal projection if and only if it is pseudo-regular, i.e. the direct sum of its isotropic part and a regular subspace.

We are going to discuss different characterizations of normal projections and a parametrization for the set of normal projections onto a (fixed) pseudo-regular subspace.

This talk is based on a joint work with A. Maestripieri.

Spectra of self-adjoint quadratic operator pencils

Manfred Möller

Universitz of the Witwatersrand
Johannesburg, South Africa

Let \(H\) be a Hilbert space and let \(A\), \(K\), \(M\) be self-adjoint operators in \(H\). We will give an overview of results on the spectral theory of the quadratic operator pencil \[L(\lambda )=\lambda ^2M-i\lambda K-A,\ \lambda \in \mathbb{C}.\] If \(K\ge0\), \(M\ge0\), are bounded, \(M+K\gg 0\) and \(A\) is bounded below with compact resolvent, then the spectrum of \(L\) is symmetric with respect to the imaginary axis and lies in the closed upper half-lane and on the imaginary axis. Particular properties of these eigenvalues will be discussed when \(K\) is a rank one operator.

Such quadratic operator pencils occur in mathematical models for problem in Mathematical Physics and Engineering, where the \(\lambda \)-quadratic term appears in the differential equation and the \(\lambda \)-linear term appears in the boundary conditions. Prime examples are are the Regge problem and a fourth order differential equation for damped vibrating beams.

A comprehensive account of the theory and applications wll be given in the forthcoming monograph "Operator Pencils and Applications" by V. Pivovarchik and M. Möller.

Bound states in PT-symmetric layers

Radek Novak

Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Czech Republik

We consider the Laplacian in a tubular neighbourhood of a hyperplane subject to non–Hermitean PT–symmetric Robin–type boundary conditions. They bring the non-self-adjointness into the problem as the probability current does not vanish on either component of the boundary and the layer therefore behaves as an open system. We analyse the influence of the perturbation in the boundary conditions on the threshold of the essential spectrum using the Birman–Schwinger principle. Our aim is to derive a sufficient condition for existence, uniqueness and reality of discrete eigenvalues. We show that discrete spectrum exists when the perturbation acts in the mean against the unperturbed boundary conditions and we are able to obtain the first term in its asymptotic expansion in a weak coupling regime.

Functions of perturbed operators

Vladimir Peller

Michigan State University, USA

I am going to speak about recent results in perturbation theory. If \(A\) is a self-adjoint operator on Hilbert space, \(f\) is a function on the real line, \(K\) is a self-adjoint perturbation of \(A\), I will consider the problem to estimate \(f(A+K)-f(A)\) in terms of \(K\). I am also going to consider various generalizations, in particular, functions of normal operators, functions of commuting \(n\)-tuples of self-adjoint operators.

Bounds on the generalized and the joint spectral radius of Hadamard products of bounded sets of positive operators on sequence spaces

Aljoša Peperko

Faculty of Mechanical Engeenering, University of Ljubljana, and
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

Recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010), Z. Huang (2011) and A.R. Schep (2011) proved inequalities between the spectral radius \(\rho\) of Hadamard product (denoted by \(\circ\)) of finite and infinite non-negative matrices that define operators on sequence spaces and the spectral radius of their ordinary matrix product. We extend these results to the generalized and the joint spectral radius of bounded sets of such operators. Moreover, we prove new inequalities even in the case of the usual spectral radius of non-negative matrices. In particular, we prove that \[\rho (A\circ B) \le \rho ^{\frac{1}{2}}((A\circ A)(B\circ B))\le \rho(AB \circ AB)^{\frac{1}{4}} \rho(BA \circ BA)^{\frac{1}{4}} \le \rho (AB)\] and \[\rho (A\circ B) \le \rho ^{\frac{1}{2}}(AB\circ BA)\le \rho(AB \circ AB)^{\frac{1}{4}} \rho(BA \circ BA)^{\frac{1}{4}} \le \rho (AB).\]

We also obtain related results in max algebra.

The work appeared in Linear Algebra and Applications.

On a Multi-Physics Coupling Mechanism

Reiner Picard

Institut für Analysis,
TU Dresden, Germany

A coupling mechanism allowing to link several linear models of various physical phenomena is presented. It is demonstrated that many problems of mathematical physics share a common structure, which is in the simplest case of the form

\[\label{eq:evo} \left(\partial_{0}M_{0}+M_{1}+A\right)U=F, \tag{$\ast$}\]

where \(\partial_{0}\) denotes the time-derivative, \(A\) is a skew-selfadjoint operator and \(M_{0}\), \(M_{1}\) are suitable bounded operators in a Hilbert space \(H\). This common structure facilitates the analysis of coupling phenomena and leads to an enlarged system of the same shape as \eqref{eq:evo}. The complexity of the coupled phenomena appears as encoded in the corresponding aggregated material law \[M\left(\partial_{0}^{-1}\right)=M_{0}+\partial_{0}^{-1}M_{1}.\] The usefulness of this structural perspective is illustrated by applications to various coupled systems.

[1] R. Picard and D. F. McGhee. Partial Differential Equations: A unified Hilbert Space Approach, volume 55 of De Gruyter Expositions in Mathematics. De Gruyter. Berlin, New York. 518 p., 2011.

[2] R. Picard. Mother Operators and their Descendants, Technical report, TU Dresden, arXiv:1203.6762v2, 2012. In press: JMAA (2013), doi:10.1016/j.jmaa2013.02.004

Quantum graph of three symmetrically coupled strips: spectral problem

Anton I. Popov

Department of Higher Mathematics, St. Petersburg National Research University of Information Technologies, Mechanics and Optics,
St.Petersburg, Russia

The quantum graph on graphene-like branching strips is considered. The Hamiltonian is determined as free 1D Schrödinger operator on each edge and some “boundary" conditions at each vertex. We obtain the conditions ensuring the point spectrum’s existence of the Schrödinger operator of the system and relations that give us the eigenvalues.

Spectral problem for a model of the Stokes flow through network

Igor Popov

Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, Russia

1D Stokes flow is considered. In the case when the viscosity and the density vary one 1D Schrodinger equation. The model of Stokes graph is suggested. It is analogous to the well-known model of quantum graph. Studying of the flow through a network is replaced by the conside3ration of the corresponding operator for the graph. Coupling conditions at vertices is derived from the continuity equation. It is analogous to the Kircchoff condition for the quantum graph. Spectral problem for the graph allows one to reveal some peculiarities of the Stokes flow.

Markovian Extensions of Symmetric Second Order Elliptic Differential Operators

Andrea Posilicano

Universita’ dell’Insubria, Como, Italy

Let \(\Omega\subset \mathbb{R}^n\) be bounded with a smooth boundary \(\Gamma\) and let \(S\) be the symmetric operator in \(L^2(\Omega)\) given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self–adjoint extensions of \(S\) by providing an explicit one–to–one correspondence between such extensions and the class of Dirichlet forms in \(L^2(\Gamma)\) which are additively decomposable by the bilinear form of the Dirichlet–to–Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Krein’s formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell–type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.

On the structure and properties of the solutions of abstract integrodifferential equations in Hilbert space

Nadezda Rautian

Lomonosov Moscow State University, Russia

We obtain and study the representations of the solutions of abstract integrodifferential equations in Hilbert space describing the process of heat propagation in media with memory, process of sound conduction in viscoelastic media etc. These representations are based on the spectral analysis of the operator-functions which are the symbols of these equations.

Spectral properties of Schrödinger operators on graphs

Jonathan Rohleder

TU Graz, Austria

In this talk Schrödinger operators on noncompact finite graphs are considered. Its spectral properties are investigated by means of an associated Titchmarsh-Weyl function.

Discrete Dirac system: rectangular Weyl functions, direct and inverse problems

Inna Roitberg

University of Leipzig, Germany

We consider a discrete Dirac-type (or simply Dirac) system:

\[\label{0.1} y_{k+1}(z)=(I_m+ \mathrm{i} z j C_k) y_k(z) \quad \left( k \in \mathbb{N}_0 \right), \tag{$\ast$}\]

where \(\mathbb{N}_0\) stands for the set of non-negative integer numbers, \(I_m\) is the \(m \times m\) identity matrix, \("\mathrm{i}"\) is the imaginary unit (\(\mathrm{i}^2=-1\)) and the \(m \times m\) matrices \(\{C_k\}\) are positive and \(j\)-unitary, that is,

\[\nonumber C_k>0, \quad C_k j C_k=j, \quad j: = \left[ \begin{array}{cc} I_{m_1} & 0 \\ 0 & -I_{m_2} \end{array} \right] \quad (m_1+m_2=m, \, \, m_1, \, m_2 \not= 0).\]

We prove the equivalence between this system and Szegö recurrence (the coefficients of Szegö reccurence are Schur coefficients).

For system \eqref{0.1} we prove the existence of a Weyl function on the semi-axis and give a procedure to construct it (direct problem).

We construct the \(S\)-node, which corresponds to system \eqref{0.1}, and the transfer matrix function representation of the fundamental solution of this system.

Finally, we solve the inverse problem on the interval and semi-axis, that is, we prove that the matrix of the potentials \(\{C_k\}\) is uniquely recovered from Weyl function.
The results presented in this talk were obtained jointly with B. Fritzsche, B. Kirstein and A. Sakhnovich.

A Carleman estimate for elliptic second order partial differential operators

Christian Rose

Department of Mathematics
Chemnitz University of Technology, Germany

In our context, a Carleman estimate is an inequality of the form \[\lVert \phi f \rVert_{L^2 (U)} \leq C \lVert \phi \Delta f \rVert_{L^2 (U)} ,\] where \(U \subset \mathbb{R}^d\) open, \(u \in C_\mathrm{c}^\infty (U)\), \(\phi\) a suitable weight function and \(\Delta\) denotes the Laplacian. Such Carleman estimates are used to study (quantitative) unique continuation principles, which in turn find application for the study of spectral properties of Schrödinger operators.

We present some history, an application and an ongoing work on a type of a Carleman estimate. It applies to general second order elliptic differential operators with Lipshitz continuous coefficients. While there exists a wide range of Carleman estimates for elliptic operators, our aim is to give a version with an explicit weight function and quantitative control of the parameter-dependence of the constants.

A joint work with M. Tautenhahn and I. Nakić.

Eigenvalue estimates for Schrödinger operators on infinite combinatorial and metric graphs

Grigori Rozenblioum

Mathematics, Chalmers University of Technology,
Gothenburg, Sweden

We give an overview of recent results (mostly, by M. Solomyak and the speaker) on the spectral properties of Schrödinger operators on combinatorial and metric graphs containing infinitely many vertices. We demonstrate an essential difference compared with these properties for the usual Schrödinger operators.

Weyl functions of canonical and Dirac systems

Alexander Sakhnovich

University of Vienna, Austria

Fundamental solutions, Weyl functions and direct and inverse problems for canonical and Dirac systems are discussed.

Approximation of the integrated density of states on sofic groups

Christoph Schumacher

Department of Mathematics
Chemnitz University of Technology, Germany

The integrated density of states (IDS) is an important tool to study operators in solid state physics. The IDS of a self-adjoint operator assigns to each energy threshold the proportion of states with energy below the given threshold. The Pastur–Shubin trace formula identifies the IDS as the limit of the IDS of matrices approximating the operator in a suitable sense. Classically, the Pastur–Shubin trace formula has been established for solids whose geometry is given by amenable groups, e.g. finite-dimensional integer lattices. We prove the Pastur–Shubin trace formula for operators which describe solids whose geometry is given by Cayley graphs of finitely generated sofic groups. Sofic groups were introduced by Gromov (1999) and Weiss (2000) and include amenable groups and residually finite groups, in particular free groups, for which the Cayley graph is a Bethe lattice. In fact, there is no group which is known to be not sofic. The operators we treat can be deterministic or random and bounded or unbounded.

Joint work with Fabian Schwarzenberger.

Uniform existence of the integrated density of states on metric Cayley graphs

Christian Seifert

Hamburg University of Technology, Germany

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 and the integrated density of states as the limit can be expressed by a Pastur–Shubin formula.

This is joint work with Felix Pogorzelski (Friedrich Schiller University Jena, Germany) and Fabian Schwarzenberger (Chemnitz University of Technology, Germany).

Accurate eigenvalue decomposition of real symmetric arrowhead and diagonal–plus–rank–one matrices and applications

Ivan Slapničar

Faculty of Electrical Eng., Mechanical Eng., and Naval Arch.
University of Split, Croatia

We present new algorithms for solving eigenvalue problems for \(n\times n\) real symmetric arrowhead matrices and rank–one modifications of diagonal matrices. The algorithms are forward stable and compute all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in \(O(n^{2})\) operations. The algorithms are based on a shift–and–invert approach. Only a single element of the inverse of the respective shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithms adaptable for parallel computing. Our results extend to Hermitian matrices, singular value decompositions, and Takagi factorizations of complex matrices. The methods can be used as a part of divide–and–conquer methods for tridiagonal problems.

This is a joint work with Nevena Jakovčević Stor and Jesse L. Barlow.

Compactness properties of Schrödinger type operators

Peter Stollmann

Department of Mathematics
Chemnitz University of Technology, Germany

We give an overview over some results on compactness of of operators. In particular, we will be dealing with the question under what circumstances resolvents and semigroups of Schrödinger type operators are compact. Such results have played an important role in various areas of Analysis and Mathematical Physics.

The characteristic function for infinite Jacobi matrices, its logarithm, the spectral zeta function, and solvable examples

Pavel Stovicek

Department of Mathematics, Faculty of Nuclear Science
Czech Technical University in Prague, Czech Republic

A function \(\mathfrak{F}\) with simple and nice algebraic properties is defined on a subset of the space of all complex sequences. In terms of \(\mathfrak{F}\) one can express the characteristic function for a certain class of infinite Jacobi matrices with discrete spectra as an analytic function on a suitable complex domain. One can show that the zero set of the characteristic function actually coincides with the point spectrum of the corresponding Jacobi matrix. As an illustration several solvable examples are discussed such that the characteristic function is explicitly expressible in terms of special functions.

Further we present a formula for the logarithm of \(\mathfrak{F}\). Using this formula and considering Jacobi matrices with a finite Hilbert-Schmidt norm and vanishing diagonal we can express the spectral zeta function as a power series in the matrix entries. In particular cases this relation gives the Rayleigh function associated with the Bessel functions and the \(q\)–Airy function.

The contribution is partially based on the papers F. Stampach, P. Stovicek: Linear Alg. Appl. 434 (2011) 1336-1353 and Linear Alg. Appl. 438 (2013) 4130-4155.

Keldysh type operators

Liudmila Suhocheva

Department of Mathematics, Voronezh State University, Russia

Let \(H\) be a Hilbert space with the scalar product \((\cdot,\cdot)\). Let \(A\) and \(B\) be compact selfadjoint operators acting in this space and \(A > 0\). Consider the Keldysh type operator \(F = A(I + B)\) and introduce in \(H\) an inner product \([\cdot,\cdot]\) given by \([\cdot,\cdot] = ((I + B)\cdot,\cdot)\) Denote \(K =\{ H , [\cdot,\cdot] \}\) . Then \(K\) is an almost Krein space, moreover, \(K\) is an almost Pontryagin space.

The aim is to study the completeness and basicity problems for selfadjoint operator F in almost Krein space and prove criteria for the basicity and completeness of root vectors of linear pencils.

Adjacency preserving maps

Peter Šemrl

Faculty of Mathematics and Physics
University of Ljubljana, Slovenia

We will present several recent structural results for adjacency preserving maps on matrix spaces and operator spaces. Applications, especially those in mathematical physics, will be discussed.

Spectral properties of discrete alloy-type models

Martin Tautenhahn

Department of Mathematics
Chemnitz University of Technology, Germany

The discrete alloy-type model is given by the family of random Schrödinger operators \(H_\omega = -\Delta + \lambda V_\omega\), \(\omega \in \Omega = \times_{k \in \mathbb{Z}^d} \mathbb{R}\), on \(\ell^2 (\mathbb{Z}^d)\), where \(\lambda > 0\) models the strength of the disorder, \(\Delta\) denotes the discrete Laplace operator and \(V_\omega\) a multiplication operator by the function \[V_\omega (x) = \sum_{k \in \mathbb{Z}^d} \omega_k u(x-x).\] We assume that the single-site potential \(u : \mathbb{Z}^d \to \mathbb{R}\) is summable and that \(\omega_k\), \(k \in \mathbb{Z}^d\), is a sequence of independent identically distributed random variables, i.e. the product space \(\Omega\) is equipped with a product measure \(\mathbb{P} = \prod_{k \in \mathbb{Z}^d} \mu\) where \(\mu\) is some probability measure on \(\mathbb{R}\).

Physically it is believed that the presence of disorder results in the fact that solutions of the Schrödinger equations stay trapped in a finite region of space for all time. This manifests mathematically in the sense that there exists intervals with only pure point spectrum; a phenomena called Anderson localization or spectral localization.

In this talk we discuss recent localization results for the discrete alloy-type model under suitable conditions on \(u\) and the measure \(\mu\). In particular, we allow the single-site potential \(u\) to change its sign. Moreover, we discuss new results on Poisson statistics, i.e. that the random process given by the (suitable scaled) eigenvalues converges to a Poisson process. While the class of potentials for which we obtain Poisson statistics is quite restricted, it is the first rigorous result where the random variables couple to a perturbation (given in terms of \(u\)) which is not of rank one.

A joint work with A. Elgart, N. Peyerimhoff and I. Veselić.

Monodromy matrices of 1d differential operator of order \(4\)

Vadim Tkachenko

Ben-Gurion University of the Negev, Beer-Sheva, Israel

Let \(p(x)\) and \(q(x),\ x\in[0,\pi]\), be a pair of real-valued functions satisfying conditions

\[\label{1} \int\limits_0^\pi (|p'(x)|^2+|q(x)|^2)dx<\infty, \tag{$\ast$}\]

and let \(\mathcal L\) be the differential operator

\[\label{2} {\mathcal L}=\frac{d^4}{dx^4} +\frac{d}{dx}\ p(x)\frac{d}{dx}+q(x),\quad x\in(0,\pi). \tag{$\ast\ast$}\]

We denote by \(U(x,\lambda), \lambda\in\mathbb C,\) the fundamental matrix of the equation \({\mathcal L}u=\lambda u\), i.e., we set \[U(x,\lambda)=||u^{(j-1)}_{k}(x,\lambda)||_{k,j=1}^4,\ U(0,\lambda)=I,\] with \(u_k(x,\lambda), k=1,...,4,\) being solutions to the above equation. It is well known that the monodromy matrix \(U(\pi,\lambda)\) contains, one way or another, all data related to the boundary problems generated by \(L\) in the interval \([0,\pi]\).

We describe the set of all \(4\times4\) matrices \(U(\lambda), \lambda\in\mathbb C,\) which are the monodromy matrices of operators \eqref{2} restricted by condition \eqref{1}.

The main tool to obtain such a description is the transformation operator introduced by Z. Leibenzon [1,2].

[1] Z. Leibenzon, Trudy Mosc. Math. Ob., Trans. MMS, 1966, v.15, 78–163.

[2] Z. Leibenzon, Trudy Mosc. Math. Ob., Trans. MMS, 1971, v.25, 13–61.

Well-posedness for a class of non–autonomous differential inclusions

Sascha Trostorff

TU Dresden, Germany

We provide a solution theory for a class of non–autonomous differential inclusions in a suitable Hilbert space setting, covering hyperbolic– and parabolic–type problems. The main idea for proving the well–posedness is to establish the operators involved as maximal monotone operators in time and space and then to apply perturbation results. The result is exemplified by the equations of visco-plasticity with time dependent coefficients.

This is a joint work with Maria Wehowski (TU Dresden).

Krein space methods for indefinite Sturm-Liouville operators

Carsten Trunk

Institute for Mathematics, TU Ilmenau, Germany

We consider Sturm-Liouville operators with an indefinite weight, i.e. operators of the form

\[\label{A} A=\frac{1}{r} \left(-\frac{d^2}{dx^2}+q \right) \tag{$\ast$} \]

on a real interval. The potential \(q\) and the weight \(r\) are real-valued, locally integrable functions, \(r\ne 0\) a.e.

Closely related to the operator \(A\) in \eqref{A} is the definite Sturm-Liouville operator

\[ B= \frac{1}{|r|}\left( -\frac{d^2}{dx^2}+q\right).\]

We have \(A=\) sgn\(\,(r) B\), and, hence, \(B\) is a selfadjoint operator in a \(L^2\)-Hilbert space if and only if \(A\) is selfadjoint in a \(L^2\)-Krein space, where sgn\(\,r\) is taken as a fundamental symmetry.

We will review some of the well-known results and then we will focus on the singular case. Our aim is to describe the spectrum of the indefinite Sturm-Liouville operator \(A\) according to the location of the spectrum \(\sigma(B)\) of the definite Sturm-Liouville operator \(B\).

Spectral analysis and correct solvability of abstract integrodifferential equations in Hilbert space

Victor Vlasov

Lomonosov Moscow State University, Russia

We study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media. The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.

Perturbation of accretive operators in \(L_p\) and elliptic operators with complex coefficients

Hendrik Vogt

Technische Universität Hamburg-Harburg, Germany

We present a new perturbation theorem for accretive operators in \(L_p\). This result is used to associate the generator of a \(C_0\)-semigroup on \(L_p(\Omega)\) with the formal differential expression \[\mathcal{L} = \nabla\cdot(a\nabla) - b_1\cdot\nabla - \nabla \cdot b_2 - Q\] on an open set \(\Omega\subseteq \mathbb{R}^N\), with complex measurable coefficients \(a\colon\Omega\to \mathbb{C}^{N\times N}\), \(b_1,b_2 \colon \Omega\to \mathbb{C}^N\) und \(Q \colon \Omega\to \mathbb{C}\).

We are particularly interested in the case that the sesquilinear form associated with \(\mathcal{L}\) is not sectorial. Then it can happen that one does not obtain a \(C_0\)-semigroup on \(L_2(\Omega)\), but only on \(L_p(\Omega)\) for certain \(p\ne2\).

Joint work with T. ter Elst, Z. Sobol and V. Liskevich.

The Porous Medium Equation as a Gradient System

Jürgen Voigt

Fachrichtung Mathematik, Technische Universität Dresden, Germany

It is shown that the Porous Medium Equation \(\dot u-\Delta u^m=f\), with \(m\in(0,\infty)\), can be modeled as a gradient system in the Hilbert space \(H^{-1}(\Omega)\), and existence and uniqueness of solutions are obtained in this framework. We deal with bounded and certain unbounded open sets \(\Omega\subseteq\mathbb{R}^n\) and do not require any boundary regularity.

This is a report on joint work with Samuel Littig.

Reference: S. Littig, J. Voigt, Porous Medium Equation and Fast Diffusion Equation as Gradient Systems. Czechoslovak Mathematical Journal, to appear.

From the color of the stars to semiclassical spectral estimates

Timo Weidl

Stuttgart University, Germany

I discuss various classical results and some recent developments on semiclassical spectral bounds on the eigenvalues of the Dirichlet Laplacian with sharp constants in the first order and additional remainder estimates of lower order.

A strange phenomenon for the singular values of commutators with rank one matrices

David Wenzel

Department of Mathematics
Chemnitz University of Technology, Germany

Motivated by finding an explanation for the typically very small difference of \(XY\) and \(YX\) we will investigate how the singular values of \(XY-YX\) may look like. Here, \(X\) and \(Y\) are certain square matrices and, in this early approach, one of them has rank one. Due to this, there are only two non-trivial numbers among the commutator’s singular values, whence the pairs of interest can be depicted in the plane. The emphasis of the analysis will lie on the unexpectedly intriguing case in which both matrices are of rank one, because the result then is surprisingly odd, and the problem gives rise to interpretations unveiling geometry acting in the background.

Spectral information contained in Dirichlet-to-Neumann-type maps

Ian Wood

University of Kent at Canterbury, UK

A useful tool in studying forward and inverse problems for ODEs is given by the Weyl-Titchmarsh m-function. In PDE problems, a similar role is played by the Dirichlet-to-Neumann map. Both of these can be determined solely from the boundary behaviour of solutions. In this talk, we will look at extending m-functions and Dirichlet-to-Neumann maps to the abstract setting of boundary triples, giving rise to operator M-functions. We will discuss properties of M-functions, their relation to the resolvent and the spectrum of the associated operator and connections to the extension theory of operators. Our focus will be particularly on non-selfadjoint cases.

SMP matrices

Peter Yuditskii

J. Kepler University of Linz, Austria

Jacobi matrices probably are the most classical object in the Spectral Theory. CMV matrices are comparably fresh one, although they are related to the very classical topic, that is, to Orthogonal Polynomials on the unit circle (in the same way like Jacobi matrices are related to Orthogonal Polynomials on the real axis). We will discuss the third member of this family. Our matrices are generated by the orthonormal systems of functions related to the so called Strong Moment Problem. For this reason we call them SMP matrices. For example, we can describe the spectral sets of periodic SMP matrices. Similarly to their twins, the description is given by means conformal mappings on hyperbolic, in this case, comb domains. We represent functional models associated with periodic and almost periodic SMP matrices. We are especially enthusiastic with respect to the role, which such matrices can play in the Killip-Simon problem related to Jacobi matrices with the essential spectrum on two arbitrary intervals.

Supported by the Austrian Science Fund FWF, project no: P22025–N18.

Weyl–Titchmarsh theory for discrete symplectic systems

Petr Zemánek

Department of Mathematics and Statistics, Faculty of Science
Masaryk University, Czech Republik

In this talk, we present the Weyl-Titchmarsh theory for discrete symplectic systems with general linear dependence on the spectral parameter. We generalize and complete several recent results concerning these systems, which have the spectral parameter only in the second equation. We characterize the Weyl disks and Weyl circles including their limiting behavior and provide an analysis of square summable solutions. Moreover, the Weyl-Titchmarsh theory for discrete symplectic systems with general jointly varying endpoints is also discussed.

Oscillations of large eigenvalues for the Jaynes–Cummings model in the rotating wave approximation

Lech Zielinski

Laboratoire de Mathematiques Pures et Appliques, Maison de la Recherche Blaise Pascal
Universite du Littoral Cote d’Opale, Calais, France

The Jaynes-Cummings model plays an important theoretical and experimental role in Quantum Optics. It describes the system of a two–level atom interacting with a quantized mode of an optical cavity and the Hamiltonian in the rotating wave approximation can be expressed as a self-adjoint operator in \(l^2\) defined by a Jacobi matrix. This talk presents a rigorous result on the behaviour of large eigenvalues of this model obtained in collaboration with Anne Boutet de Monvel (Université Paris 7). We show that the \(n\)-th eigenvalue can be expressed in the form \(\lambda_n =c_1n+c_0+n^{-1/4}r(n)\) where \(c_1\), \(c_0\) are constants and \(r(n)\) is bounded. Moreover we prove an asymptotic formula for \(r(n)\) revealing oscillations and the form of oscillations determines the values of parameters used in the model, i.e. the inverse problems is solved. A similar asymptotic formula was considered for two-level models e.g. by I. D. Ferenchuk, L. I. Komarov, A. P. Ulyanenkov [J. Phys. A: Math. Gen. 29 (1996) 4035-4047]. Our proof is based on an approximation of the Schrödinger evolution, i.e. the Fourier transform of the spectral mesure.

Spectral asymptotics of self-adjoint fourth order boundary value problems with eigenvalue parameter dependent boundary conditions

Bertin Zinsou

University of the Witwatersrand,
Johannesburg, South Africa

A regular fourth order differential equation with \(\lambda\)-dependent boundary conditions is considered. For four distinct cases with exactly one \(\lambda\)-independent boundary conditions, the asymptotic eigenvalue distribution is presented.