# Program

## Thursday, May 18

For participants arriving on Thursday we can meet for dinner at the Hofbräuhaus in Harburg.

## Friday, May 19

09:00-09:30

Form inequalities for symmetric L∞-contractive semigroups

TBA

09:30-10:00

Intermediate Spaces and Perturbations of Bi-Continuous Semigroups

We present a construction of intermediate and extrapolation spaces for bi-continuous semigroups. One motivation is the Desch-Schappacher perturbation theorem, which uses these spaces. A possible extension of this perturbation result will be also presented.

10:00-10:30

On the domain of fractional powers of operators in Banach spaces and generalised Dirichlet-to-Neumann maps

Let $A: \mathcal{D}(A) \rightarrow X$ be a non-negative linear operator in a Banach space $X$ and $\alpha \in \mathbb{C}$, $0 < \operatorname{Re} \alpha < 1$.
We study the generalised Dirichlet-to-Neumann map for the Bessel-type differential equation
\begin{equation*}
u''(t) + \frac{1-2\alpha}{t}u'(t) = Au(t), \quad u(0) = x.
\end{equation*}
As it turns out, for ``nice'' $x$, the generalised Neumann data $\lim\limits_{t \rightarrow 0+} -t^{1-2\alpha} u'(t)$
equals $A^\alpha x$ up to a constant $c_{\alpha} \in \mathbb{C}$.

10:30-11:00

Coffee break

11:00-11:30

Well-posednes of infinite-dimensional networks of hyperbolic partial differential equations

We consider the well-posedness of a class of hyperbolic partial differential equations on a one dimenional spatial domain. This class includes in particular infinite-dimensional networks of transport, wave and beam equations. Equivalent conditions for contraction semigroup generation are derived. As an application we consider partial differential equations on the semiaxis.

11:30-12:00

Infinite-dimensional input-to-state stability and Orlicz spaces

In this talk, the relation between input-to-state stability and integral input-to-state stability is studied for linear infinite-dimensional systems with an unbounded control operator. Although a special focus is laid on the case $L^{\infty}$, general function spaces are considered for the inputs. We show that integral input-to-state stability can be characterized in terms of input-to-state stability with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability notions with respect to $L^\infty$ are equivalent.

This talk is based on a joint work with Birgit Jacob, Jonathan R. Partington, and Felix L. Schwenninger.

This talk is based on a joint work with Birgit Jacob, Jonathan R. Partington, and Felix L. Schwenninger.

12:00-12:30

Hamiltonian partial differential equations on a semi-axis

We consider the partial differential equation
\begin{equation*}
\begin{array}{rcl}
{\textstyle\frac{\partial x}{\partial t}}(\zeta,t)&=&\bigl(P_1{\textstyle\frac{\partial}{\partial \zeta}}+P_0\bigr)\bigl(\mathcal{H}(\zeta)x(\zeta,t)\bigr)
\\
x(\zeta,0)&=&x_0(\zeta)
\end{array}
\end{equation*}
where $P_1\in\mathbb{C}^{d\times d}$ is Hermitian and invertible, $P_0\in\mathbb{C}^{d\times d}$ is arbitrary, and $\mathcal{H}=\mathcal{H}(\zeta)$ is positive. If $\zeta\in[a,b]$ and $\mathcal{H}$ is bounded and bounded away from zero, then there is an extensive theory of the above class of equations adressing, e.g., $\text{C}_0$-semigroup generation via boundary conditions and corresponding stability properties. In the talk we will discuss the situation where $\zeta\in[0,\infty)$.

12:30-14:00

Lunch

14:00-14:30

On Dirac operators with electrostatic δ-shell interactions

The Dirac operator is the relativistic counterpart of the Schr\"odinger operator and hence, it appears in many applications in quantum mechanics.

In this talk I will discuss the Dirac operator with an electrostatic $\delta$-shell interaction which is formally given by $A_\eta := A_0 + \eta \delta_\Sigma I_4$, where $A_0$ is the free Dirac operator, $I_4$ is the $4 \times 4$ identity matrix, $\eta \in \mathbb{R}$ and $\Sigma$ is the boundary of a smooth domain. After establishing the self-adjointness of $A_\eta$, it turns out that some of the spectral properties of $A_\eta$ are of a different nature, if $\eta = 2 c$, where $c$ denotes the speed of light, or $\eta \neq 2 c$. In the latter case I discuss finiteness of the discrete spectrum and the existence and completeness of the wave operators for the pair $\{ A_\eta, A_0\}$ and I compute the nonrelativistic limit of $A_\eta$, as the speed of light $c$ tends to $\infty$. In the case $\eta = 2 c$ these properties either differ from those for $\eta \neq 2 c$ or they are still unknown.

This talk is based on joint works with J. Behrndt, P. Exner, and V. Lotoreichik.

In this talk I will discuss the Dirac operator with an electrostatic $\delta$-shell interaction which is formally given by $A_\eta := A_0 + \eta \delta_\Sigma I_4$, where $A_0$ is the free Dirac operator, $I_4$ is the $4 \times 4$ identity matrix, $\eta \in \mathbb{R}$ and $\Sigma$ is the boundary of a smooth domain. After establishing the self-adjointness of $A_\eta$, it turns out that some of the spectral properties of $A_\eta$ are of a different nature, if $\eta = 2 c$, where $c$ denotes the speed of light, or $\eta \neq 2 c$. In the latter case I discuss finiteness of the discrete spectrum and the existence and completeness of the wave operators for the pair $\{ A_\eta, A_0\}$ and I compute the nonrelativistic limit of $A_\eta$, as the speed of light $c$ tends to $\infty$. In the case $\eta = 2 c$ these properties either differ from those for $\eta \neq 2 c$ or they are still unknown.

This talk is based on joint works with J. Behrndt, P. Exner, and V. Lotoreichik.

14:30-15:00

An Interface Coupling Problem Between Electrodynamics and Elasticity

We consider the problem of coupling electrodynamic effects with elastic properties via interaction on a common interface. The resulting transmission problem is formulated in the framework of evo-systems (i.e. “evolutionary equations”). Natural transmission conditions are derived and dynamic well-posedness of the resulting evo-system is shown.

15:00-15:30

Spectral Singularities and the Spectral Expansion for the Non-self-adjoint Schrodinger Operator with a Periodic Potential

I am going to give a talk about the construction of the spectral expansion of the one dimensional Schrödinger operator L(q) acting in L₂(-∞,∞), where q is an arbitrary complex-valued locally integrable and periodic potential. In other word, we consider in detail the spectral expansion for the general case when the operator L(q) is not a spectral operator. Note that the construction of the complete spectral decomposition appears to have been open for about 50 years. To give a complete spectral decomposition we introduce new concepts as essential spectral singularities and singular quasimomenta.

15:30-16:00

Coffee break

16:00-16:30

A Limit-Point- and Limit-Circle Classification for PT-symmetric operators

We consider a second-order differential equation
\begin{align}\label{DE}
-y''+q(x)y(x)=\lambda y(x)
\end{align}
with complex-valued potential $q$ and eigenvalue parameter $\lambda$. In $\mathcal{PT}$-symmetric quantum mechanics $x$ is on a contour $\Gamma \subset \mathbb{C}$. If the contour $\Gamma$ is chosen in a very simple way, $\Gamma:=\left\{xe^{i\phi \mathrm{sgn} x}:x \in \mathbb{R}\right\}$, then the above problem splits into two differential equations on the semi-axis $[0,\infty)$ and on $(-\infty,0]$, respectively. We provide a limit-point/limit-circle-classification of this problem and a first (rough) estimate for the spectrum of the semi-axis operators. Moreover, via boundary conditions at zero, we associate with \eqref{DE} a full line operator which is a one-dimensional perturbation of the direct sum of the semi-axis operators. We characterize all boundary conditions at zero such that the corresponding full line operator is selfadjoint in a Krein space or $\mathcal{PT}$-symmetric. A key result of our investigation is the following: The resolvent set of the full line operator associate to \eqref{DE} is non-empty and the spectrum consists of isolated eigenvalues which accumulate to infinity only.

16:30-17:00

Locally finite extensions of direct sum operators

We consider the infinite direct sum of symmetric operators. In general there is no natural boundary triplet for the adjoint even if there is one for every summand operator. We study locally finite extensions, a subclass of extensions, which can be described in terms of the summand boundary triplets and investigate the self-adjointness, the semi-boundedness from below and the discreteness of the spectrum. Sufficient conditions for these properties are obtained from the theory of weighted discrete Laplacians. The results are then used to study Laplacians on metric graphs with a countable (possibly infinite) set of vertices and edges. In particular, we allow graphs with arbitrarily small edge length and study the spectral properties of point-interactions on these graphs.

This talk is based on a joint work with Carsten Trunk (TU Ilmenau).

This talk is based on a joint work with Carsten Trunk (TU Ilmenau).

17:00-17:30

Stochastic Evolutionary Equations

We present an extension of the theory of evolutionary equations, that is, integro-differential-algebraic partial differential equations involving time in the sense developed by Picard in 2009, to a stochastic setting. The setting allows for showing well-posedness results for classical examples of the wave or heat equation with multiplicative noise within one single approach. Moreover, the method paths the way to incorporating Maxwell's equations or equations changing its type from parabolic to hyperbolic to elliptic type on different spatial domains.

This is joint work with A. Süß.

This is joint work with A. Süß.

19:00

Conference Dinner

## Saturday, May 20

09:00-09:30

Friedlander's inequality for quantum graphs

We discuss to what extent Friedlander's inequality between Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain can be extended to the Laplacian on a finite metric graph.

09:30-10:00

Eigenvalue inequalities for the Laplacian with mixed boundary conditions

The Laplacian with mixed boundary conditions arises e.g. in the study of nodal domains for Neumann eigenfunctions and in the construction of isospectral domains. In this talk, we will discuss inequalities for the eigenvalues of the Laplacian subject to mixed boundary conditions on a bounded Lipschitz domain. Particular attention will be paid to convex polyhedral domains. We estimate the eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of it in terms of either Dirichlet or Neumann eigenvalues. These results complement and generalize several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and Filonov. The proofs rely on the min-max principle and follow the strategy of the papers by Filonov and by Levine and Weinberger with new geometric insights involved.

These results are obtained in collaboration with Jonathan Rohleder.

These results are obtained in collaboration with Jonathan Rohleder.

10:00-10:30

Parseval frames for Krein spaces

Along this talk we briefly recall the notion of $J$-frames for Krein spaces. Then, we present a particular class of $J$-frames, the so-called Parseval $J$-frames. After studying this class, we describe Parseval $J$-frames as the vectors obtained by applying selfadjoint projections (in the Krein space sense) to orthonormal bases of a larger Krein space. This is a Krein space version of the well-known Naimark's Theorem for frames in Hilbert spaces.

10:30-11:00

Coffee break

11:00-11:30

Dynamical Sampling

Classical sampling theory deals with the recovery of functions from evaluations (samples). For this, it is clear that the sampling grid has to consist of sufficiently many points. In a real world scenario these points may represent sensors which measure, e.g., temperature or air pollution. In Dynamical Sampling one seeks to use less sensors by exploiting the dynamics of the signals to be sampled. The mathematical task can be formulated as follows:

In the presented work we elaborate on the question which normal operators $A$ and finite families $F$ allow for Dynamical Sampling. The result is an interesting mixture of frame theory, spectral theory, and interpolation theory on Hardy spaces. We also give conditions on non-normal operators $A$. It turns out that it is necessary that $A$ be similar to an operator belonging to a special class of contractions.

The talk is based on joint work with C. Cabrelli, U. Molter, and V. Paternostro (all from Universidad de Buenos Aires).

*Given a bounded operator $A$ (the evolution operator) in a Hilbert space $H$ (the signal class) and a countable family $F$ of vectors in $H$ (representing the places), when is the set of iterations $$ \{A^{n}f : n\in\mathbb N,\,f\in F\} $$ a frame for $H$?*In the presented work we elaborate on the question which normal operators $A$ and finite families $F$ allow for Dynamical Sampling. The result is an interesting mixture of frame theory, spectral theory, and interpolation theory on Hardy spaces. We also give conditions on non-normal operators $A$. It turns out that it is necessary that $A$ be similar to an operator belonging to a special class of contractions.

The talk is based on joint work with C. Cabrelli, U. Molter, and V. Paternostro (all from Universidad de Buenos Aires).

11:45

General Assembly of the GAMM Activity Group

In the afternoon we offer an excursion to the City of Hamburg, including a visit of the Plaza of new Elbphilharmonie