Conference schedule
Morning session (congress hall Mimoza II)
Monday | Tuesday | Wednesday | Thursday | Friday | |
8:50 - 9:00 | Conference opening | ||||
9:00 - 10:00 | Fröhlich | Exner | Hackbusch | Böttcher | Boulton |
10:00 - 11:00 | Griesemer | Tretter | Poster Session (summary) | Langer | Drmač |
11:00 - 11:30 | Coffee break | Coffee break | Coffee break | Coffee break & Poster session (discussion) | Coffee break |
11:30 - 12:00 | Truhar | Vogt | Motovilov | Kaimanovich | |
12:00 - 12:30 | Picard | Antonić | Rehberg | Komech/Kopylova | |
12:30 - 14:00 | Lunch break | Lunch break | Lunch break | Lunch break | Lunch break |
Afternoon session (congress hall Mimoza II)
Monday | Tuesday | Wednesday | Thursday | Friday | |
14:00 - 14:30 | Gilbert | Hansmann | Excursion & Conference dinner | Šemrl | Popov |
14:30 - 15:00 | Hanauska | Khrabustovskyi | Plestenjak | Täufer | |
15:00 - 15:30 | Giribet | Gernandt | Kalaj | Wyss | |
15:30 - 16:00 | Coffee break | Coffee break | Coffee break | Coffee break & closing of the conference | |
16:00 - 16:30 | Belyi | Rohleder | Wood | ||
16:30 - 17:00 | Mitrović | Kühn | Holzmann | ||
17:00 - 17:30 | Waurick | Currie | Schumacher |
Afternoon session (congress hall Mimoza I)
Monday | Tuesday | Wednesday | Thursday | Friday | |
14:00 - 14:30 | Fischbacher | Košir | Excursion & Conference dinner | Kondej | |
14:30 - 15:00 | Tautenhahn | Šepitka | Kostenko | Judge | |
15:00 - 15:30 | Vlasov/Rautian | Stoiciu | Eckhardt | Peperko | |
15:30 - 16:00 | Coffee break | Coffee break | Coffee break | Coffee break & closing of the conference | |
16:00 - 16:30 | Simon Hilscher | Burazin | Tambača | ||
16:30 - 17:00 | Trostorff | Lazar | Sakhnovich | ||
17:00 - 17:30 | Martínez Pería | Novak | Torshage |
Poster sessions
Each poster will be assigned to a place in the lobby of the conference center. The lobby area in front of the Mimosa lecture hall will be reserved for the conference during the whole week. The space available to each poster is equivalent to an area of DIN A0 (corresponds to 84,1 x 118,9 cm) and will be labeled with the name of the poster presenter. The posters should be attached with a nonaggressive substance such as bluetack or scotch tape. Sufficient amount will be provided. The available poster areas will be labeled from Monday morning, so poster presenters will be able to display their posters from Monday afternoon onwards. The posters can be left on show during the entire meeting and should be removed by Friday morning.
The poster section will begin with a plenary summary session on Wednesday morning followed by the poster presentation and the discussion session on Thursday. Each poster presenter will have an opportunity to give a 3-minute summary of his/her poster. No questions will be asked during the presentation. The summaries will be presented in the Mimosa 2 lecture hall. Please note that the time limit for the summary will be strictly enforced.
Each poster presenter may use up to three slides for the poster summary session. The presentation, as a pdf file, should be sent to luka.grubisic@math.hr before September 21, 2015. All master files will be put on the same computer for the presentation in the summary session. There will be an award for the best poster.
Abstracts
Friedrichs systems on complex spaces
Nenad Antonić
Maths Department, Faculty of Science, University of Zagreb, Croatia
Abstract:
Symmetric positive systems of first-order linear partial differential equations
were introduced by Kurt Otto Friedrichs (thus they are often called Friedrichs' systems today)
in an attempt to treat equations that change their type, like the equations modelling
transonic fluid flow.
A Friedrichs system consists of a certain first order system of partial differential equation and
an admissible boundary condition.
Friedrichs showed that this class of problems encompasses a wide variety of
classical and neoclassical initial and boundary value problems for various
linear partial differential equations.
More recently, Ern, Guermond and Caplain (2007) suggested another approach to Friedrichs' theory,
which was inspired by their interest in the numerical treatment of Friedrichs systems.
They expressed it in terms of operators acting on abstract Hilbert spaces and proved well-posedness
result in this abstract setting.
We (Antonić & Burazin, 2010) rewrote their cone formalism in terms of an indefinite inner
product space, which in a quotient by its isotropic part gives a
Krein space. This new viewpoint allowed us to show that the three
sets of intrinsic boundary conditions are actually equivalent,
which facilitates further investigation of their precise relation to the original Friedrichs boundary conditions.
Although some evolution (non-stationary) problems can be treated within this framework,
their theory is not suitable for problems like the initial-boundary value problem for
the non-stationary Maxwell system, or the Cauchy problem for the symmetric hyperbolic system.
This motivates the interest in non-stationary Friedrichs systems.
Some numerical treatment of such problems was already done by Burman, Ern and Fernandez (2010),
while the existence and uniqueness result was recently provided by Burazin and Erceg.
Most classical papers deal with Friedrichs systems in real space setting. In this talk
we shall address the extensions of both stationary and non-stationary theory to complex spaces,
as well as the two-field theory, commenting on the difficulties encountered in the semilinear case,
as well as in the Banach space setting. Finally, we shall investigate the applicability of these
extensions to some examples, like the Dirac or Maxwell system.
This is a joint work with Krešimir Burazin, Marko Erceg and Ivana Vuksanović.
Conservative L-systems and the Livsic function
Sergey Belyi
Troy University, USA
Abstract:
We study the connections among: (i) the Livšic class of functions $s(z)$ that are the characteristic functions of a densely defined symmetric operators $\dot A$ with deficiency indices $(1, 1)$; (ii) the characteristic functions $S(z)$ (the Möbius transform of $s(z)$) of a maximal dissipative extension $T$ of $\dot A$ determined by the von Neumann parameter $\kappa$; (iii) the transfer functions $W_\Theta(z)$ of conservative L-systems $\Theta$ with the main operator $T$ of the system. It is shown that under some natural hypothesis $S(z)$ and $W_\Theta(z)$ are reciprocal to each other. We establish that the impedance function of a conservative L-system coincides with the function from the Donoghue class if and only if the von Neumann parameter vanishes ($\kappa=0$). Moreover, we introduce the generalized Donoghue class and provide the criteria for an impedance function to belong to this class. We also obtain the representation of a function from this class via the Weyl-Titchmarsh function.
The talk is based on joint work with K. A. Makarov and E. Tsekanovskii (see also references below).
- S. Belyi, K. A. Makarov, E. Tsekanovskii: Conservative L-systems and the Livšic function, Methods of Functional Analysis and Topology, (2015), (to appear).
- K. A. Makarov and E.Tsekanovskii: On the Weyl-Titchmarsh and Livšic functions, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 87 (2013), 291--313.
- Yu. Arlinskii, S. Belyi, E. Tsekanovskii: Conservative Realizations of Herglotz-Nevanlinna functions, Operator Theory: Advances and Applications, Vol. 217, Birkhäuser Verlag, 2011.
Surface states and catalytic activity
Irina V. Blinova
ITMO University, St. Petersburg, Russia
Abstract:
At present, nanocatalists are the most widely used catalyst. It is related with good surface/volume ratio. One can mention an interesting phenomenon: the catalytic activity of nanocatalyst increases considerably if there are irregular inclusions at the nanoparticle surface. It can be related with the change of surface electron states. Solvable models based on the theory of self-adjoint extensions of symmetric operators are suggested. The nanoparticle is considered as a half-crystal. The model system without impurity (pure half-crystal) has purely continuous spectrum. Appearance of irregular surface impurity leads to an opportunity for the Hamiltonian to have non-empty discrete spectrum. The corresponding bound state is localized near the nanoparticle surface. It leads to increasing of the electron density near the surface which is related with the increasing of the catalytic activity. The simplest model is one-dimensional. In this case, we found the relation between the strength of the impurity and the geometry of the system which ensure the appearance of the bound state. More realistic 3D model allows one to compute the bound states (the dispersion equation is derived in an explicit form).
Numerical approximation of Poincaré - Friedrichs constants
Lyonell Boulton
Department of Mathematics, Heriot-Watt University, UK
Abstract:
The amplitude of a regular function which vanishes on the boundary of a compact region is controlled (integral square) by a constant times the gradient. The smallest possible value of the constant, allowing this to hold true for all functions in the suitable Sobolev space, is non-zero and it is often called the Poincaré - Friedrichs constant. A similar statement is still valid, if we replace the gradient by the curl operator.
Unfortunately, we cannot estimate numerically guaranteed upper bounds for Poincaré - Friedrichs constants by means of a direct application of the classical Galerkin method. The latter might lead, for example, to variational collapse in the case of the curl operator. In this talk we will examine two methods for overcoming this difficulty.
Lattice theory and Toeplitz determinants
Albrecht Böttcher
Technische Universität Chemnitz, Germany
Abstract:
A lattice is a discrete subgroup in a finite-dimensional Euclidean space.
Lattice theory has numerous applications, for instance, in discrete
optimization or coding theory. One of these applications consists in associating
lattices with elliptic or related curves over finite fields, which are
of prominent use in coding theory, and then to connect arithmetic properties
of the curve with geometric properties of the lattice. A basic quantity
of every lattice is the volume of its fundamental domains. This volume is
the determinant of some matrix, and in several interesting cases this
matrix is just a perturbed Toeplitz matrix. The generating function of
this Toeplitz matrix is in general not well-behaved, so that the classical
Szego limit theorem for Toeplitz determinants cannot be used. It rather
turns out that the generating function is often a so-called Fisher-Hartwig
symbol. As such symbols are also emerging in statistical physics,
their determinants have been thoroughly studied for decades. The results
of these studies now prove to be of use in lattice theory. The talk
is an introduction to some aspects of lattice theory and Toeplitz determinants
and also exhibits some recent results obtained in joint work with
Lenny Fukshansky, Stephan Ramon Garcia, and Hiren Maharaj.
Exact solutions in optimal design problems for stationary diffusion equation
Krešimir Burazin
Department of Mathematics, University of Osijek, Croatia
Abstract:
We consider multiple state optimal design problems for stationary diffusion in the case of two isotropic phases, aiming to minimize a linear
combination (with nonnegative coefficients) of compliances. Commonly, optimal design problems do not have classical solutions, so one considers proper relaxations of original problems. A relaxation by the homogenization method consists in introducing generalized materials, which are mixtures of original materials on
the micro-scale.
It is well known that for problems with one state equation, there exist
relaxed solutions which correspond to simple laminates at each point of
the domain. As a consequence, one can write down a simpler relaxation,
ending by a convex minimization problem.
For multiple state optimal design problems we derive analogous result
in the spherically symmetric case. To be precise, we consider the simpler
relaxation and prove that there exists optimal
design which is a radial function, and which is also solution for proper
relaxation of original problem. Moreover, we use the necessary and sufficient conditions of optimality to calculate this radial optimal design, and
demonstrate the procedure on some examples of optimal design problems,
where the presented method enables us to explicitly calculate the solution.
Joint work with Marko Vrdoljak.
Numerical analysis of the DSMC algorithm for multi-agent systems
Bojan Crnković
Department of Mathematics, University of Rijeka, Croatia
Abstract:
In this work we consider a problem of area coverage by multiple agents that can be used for planing of search-and-rescue, inspection or continuous surveillance missions. The goal of the multiple cooperative agents is to uniformly cover the considered area and detect mobile targets with known probability distribution. The dynamics of the multi-agent system is governed by Dynamic Spectral Multiscale Coverage algorithm. We analyze hypothetical scenarios and detect situations where DSMC algorithm shows undesirable behavior. The shortcomings of the algorithm are analyzed and explained.
Scattering on the line with a transfer condition
Sonja Currie
University of the Witwatersrand, South Africa
Abstract:
We consider scattering on the line with a transfer condition at the origin.
Part 1 of the talk will investigate the forward problem. In particular, showing that under suitable growth conditions on the potential the spectrum consists of a finite number of eigenvalues which are negative real numbers while the remainder is continuous spectrum which is comprised of the positive real axis. Asymptotics are provided for the Jost solutions and conditions are found which characterize transfer conditions which result in self-adjoint problems. Properties are given of the scattering coefficient linking it to the spectrum.
Part 2 of the talk will then concentrate on the corresponding inverse scattering problem. We will show that the transfer matrix can be reconstructed from
the eigenvalues and reflection coefficient. In addition, for potentials with compact essential support, we will show that the potential can be uniquely reconstructed.
3-D flow of a compressible viscous micropolar fluid with spherical symmetry: exponential stability of the solution
Ivan Dražić
University of Rijeka, Faculty of Engineering, Croatia
Abstract:
We consider the non-stationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain that is the subset of $\mathbf{R}^3$ bounded with two concentric spheres that present solid thermo-insulated walls. Under the assumption that the fluid is perfect and polytropic in the thermodynamical sense as well as that the initial density and temperature are strictly positive and that the initial data are sufficiently smooth spherically symmetric functions, the corresponding problem with homogeneous boundary data has a unique generalized solution for any time interval $\left[0,T\right]$, $T\in\mathbf{R}^+$.
In this work we prove that the generalized solution of the described problem define a nonlinear $C_0$-semigroup $S(t)$ and show that the semigroup $S(t)$ is exponentially stable.
Accurate computation of matrix spectral decompositions and applications
Zlatko Drmač
University of Zagreb, Croatia
Abstract:
Modern theoretical developments and exciting applications in applied sciences and engineering demand efficient and numerically sound algorithms for matrix computations.
Our aim is to illustrate how some recent developments in numerical linear algebra (accurate algorithms for numerical computation of eigenvalues and singular values, and corresponding theory) improve numerical computations in various applications. In particular, we stress the importance of error and perturbation analysis that identifies relevant condition numbers and guides computation with noisy data.
Proper identification of a condition number that governs the forward error under a class of perturbations is the key for the design of a robust matrix algorithm.
We will review basic ideas and use few separate topics as case studies. For instance, we will argue that it is possible, contrary to the usual statements, to compute highly accurate spectral information (e.g. singular values and vectors) of notoriously ill-conditioned Cauchy, Vandermonde and related e.g. Hankel matrices. Such highly accurate algorithms allow better solutions of problems e.g. in Adamyan-Arov-Krein and Carathéodory-Feyér rational approximation theories, rational matrix valued least squares fitting of frequency response measurements of dynamical systems. Numerical examples will be used to illustrate the power of the new algorithms.
The inverse spectral transform for the conservative Camassa-Holm flow
Jonathan Eckhardt
University of Vienna, Austria
Abstract:
By solving an inverse spectral problem for an indefinite Sturm-Liouville problem, we show that the conservative Camassa-Holm system with decaying initial data is equivalent to an (in general) infinite dimensional completely integrable Hamiltonian system. In particular, the conservative Camassa-Holm flow is linearized by suitably chosen spectral quantities.
Schrödinger operators exhibiting parameter-dependent spectral transitions
Pavel Exner
Doppler Institute for Mathematical Physics and Applied Mathematics, Prague, Czech Repub
Abstract:
This talk is concerned with several examples of Schrödinger operators with potentials which are below unbounded but their negative part is localized in narrow channels. They have the common property that they exhibit a parameter-dependent spectral transition: if the coupling constant exceeds a critical value the spectrum will cover the whole real axis, corresponding to the particle escape to infinity. A prototype of such a behavior can be found in the so-called Smilansky model. We review its properties and analyze a regular version of this model, as well as another system in which $x^py^p$ potential is amended by a negative radially symmetric term; in the latter case the subcritical spectrum is purely discrete. The results come from a common work with Diana Barseghyan and Miloš Tater.
On proper dissipative extensions
Christoph Fischbacher
University of Kent, UK
Abstract:
We study the problem of finding proper dissipative extensions of a given dissipative operator and its formal adjoint. Given a dissipative operator fulfilling certain assumptions, we discuss what is the natural choice for its formal adjoint. Then, we give a necessary and sufficient condition for a proper extension of such an operator to be dissipative. We apply our result in order to characterize all proper sectorial and accretive extensions of a given positive symmetric operator and all dissipative extensions of a symmetric operator with zero in its field of regularity and discuss the relation of these results to previous work by Grubb, Arlinskii and others in the area. After a short discussion of the complete non-self-adjointness of these extensions, we focus on symmetric operators with a weak dissipative perturbation and finally on more singular dissipative operators.
The operator-theoretic renormalization group
Jürg Fröhlich
ETH Zürich, Switzerland
Abstract:
I review the operator-theoretic renormalization group based on the Feshbach-Schur map (originally introduced by V. Bach, I. M. Sigal and myself) and apply it to the theory of resonances of atoms and molecules interacting with the quantized electromagnetic field.
Laplacians on infinite graphs
Hannes Gernandt
TU Ilmenau, Germany
Abstract:
Quantum graphs with infinitely many edges and vertices are investigated. We are interested in the properties of the Laplacians associated with the graph. Here we allow the infimum of all edge lengths to be equal to zero. In this setting the usual description of the graph Laplacians with boundary triplets is not immediately possible.
Therefore a regularization technique from Kostenko, Malamud and Neidhardt (cf. [1,2]) is used to obtain a boundary triplet. In [1] this technique was used to study point interactions on the real half-axis. Here we use this regularization to characterize all self-adjoint and semi-bounded graph Laplacians. It turns out that the Kirchhoff conditions do not lead to a self-adjoint operator, so that additional boundary conditions are necessary.
Furthermore we give sufficient conditions for the spectrum to be purely discrete. For this case we prove a Weyl-type asymptotic formula.
- A. Kostenko, M.M. Malamud: 1-D Schrödinger operators with local point interactions on a discrete set, J. Differential Equations 249, 259-304 (2010).
- M.M. Malamud, H. Neidhardt: Sturm-Liouville boundary value problems with operator potentials and unitary equivalence, J. Differential Equations 252, 5875-5922 (2012).
Reformulation of the Weyl-Kodaira eigenfunction expansions associated with the one-dimensional Schrodinger operator
Daphne J. Gilbert
Dublin Institute of technology, Ireland
Abstract:
We consider the one-dimensional Schrodinger operator $H$ on $L_{2}(-\infty, \infty)$ associated with
\begin{equation*}
Lu := - u ^{''} + q(r) u = \lambda u, - \infty < r < \infty,
\end{equation*}
where $q(r): R \rightarrow R$ is locally integrable, $\lambda \in R$ is the spectral parameter, and the differential expression $L$ is in Weyl's limit point case at both infinite endpoints. In this case the unique selfadjoint operator H is defined by
\begin{equation*}
Hf = Lf, f \in D(H),
\end{equation*}
where $f \in D(H)$ when $f, Lf \in L_{2}$, and $f, f'$, are locally absolutely continuous on $R$. We refer to $H$ as the Schrodinger operator on the line.
Starting from the well known Weyl-Kodaira expansion [1], we use some subsequent results of Kac [2], together with the more recent theory of subordinacy [3] to reformulate the the original expansion. This enables us to derive a new expansion in which many of the principal underlying
features of the original expansion, for example the existence of a scalar spectral density function and the multiplicity properties of the spectrum, are explicitly exhibited. The new formulation also enables those generalised eigenfunctions which indicate the location of the spectrum to be identified, together with their spectral types which may be absolutely continuous, singular continuous, or pure point. The main steps of the proof will be outlined , and the results illustrated by a few
simple examples (see e.g. [4]).
References:
[1] K. Kodaira, The eigenvalue problem for ordinary differential equations and Heiseberg's theory of S -matrices, Amer. J. Math. 71 (1949), 921-945.
[2] I.S. Kac, On the multiplicity of the spectrum of a second order differential operator and the associated eigenfunction expansion, Izv. Akad. Nauk SSSR
Ser. Mat. 27 (1963), 1081-1129 (in Russian).
[3] D.J. Gilbert, On subordinacy and analysis of the spectrum of one-dimensional Schrodinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh
Sect. A 128(1989), 213-229.
[4] D.J. Gilbert, Eigenfunction expansions associated with the one-dimensional Schrodinger operator, Oper. Theory: Advances Appl. 227 (2013), 89-105.
Indefinite least squares problems and pseudo-regularity
Juan Giribet
University of Buenos Aires, Argentina
Abstract:
Given two Krein spaces $\mathcal{H}$ and $\mathcal{K}$, a (bounded) closed-range operator $C : \mathcal{H} \rightarrow \mathcal{K}$ and a vector $y \in\mathcal{K}$, the indefinite least squares problem consists in finding those vectors $u \in\mathcal{H}$ such that
$$
[Cu - y,Cu - y] = \min_{x\in\mathcal{H}}[Cx - y,Cx - y].
$$
This work is devoted to give necessary and sufficient conditions for the existence of solutions of this abstract problem.
The indefinite least squares problem has been thoroughly studied in finite-dimensional spaces, but the usual assumption on the range of $C$ is too strong, say the range is a uniformly $J$-positive subspace of $\mathcal{K}$. Along this manuscript the range of $C$ is only supposed to be a $J$-nonnegative pseudo-regular subspace of $\mathcal{K}$.
Spectral Theory of the Fermi Polaron
Marcel Griesemer
University of Stuttgart, Germany
Abstract:
The experimental realization of fermionic superfluidity with imbalanced spin populations has triggered a lot of theoretical work in the past ten years on the simple model system consisting of N spin-up fermions interacting with one spin-down particle via attractive two-body delta-interactions. We discuss the self-adjoint realization of this model in 2 space dimensions and we establish a Birman-Schwinger principle for its eigenvalues. Energies computed by physicists of polaron and molecule states turn out to be upper bounds for the ground state energy, as desired. Our analysis of these energies in the limits of weak and strong coupling confirms the existence of a polaron-to-molecule transition. (Joint work with Ulrich Linden.)
Resolvent estimates for saddle point quadratic forms
Luka Grubišić
University of Zagreb, Croatia
Abstract:
We present resolvent estimates for indefinite quadratic forms of the saddle point type by means of abstract diagonalization. To this end we study form Riccati equations. Indefinite quadratic forms of the saddle point type describe dynamical properties of constrained systems of partial differential equations. As a main example of an application of our theory we consider the Stokes system and recover several known results on the spectral theory of the associated operator representation using manly geometric means such as a finer control of the rotation of reducing graph subspaces under form perturbations. We also provide results on the solvability of the form Riccati equation which might be of independent interest.
This is a joint work with V. Kostrykin, K. A. Makarov, S. Schmitz and K. Veselić.
Numerical Treatment of Tensors
Wolfgang Hackbusch
Max-Planck-Institut Mathematik in den Naturwissenschaften, Leipzig, Germany
Abstract:
In various applications large-scale tensors appear whose data size is far
beyond the capacity of any computer. Nevertheless, tensors can be treated
numerically, if they can be approximated by appropriate sparse
representation formats. The classical formats are the canonical format and
the Tucker format. The first one leads to numerical and theoretical
difficulties, since the set of tensors of rank $\leq r$ is not closed. The
latter format works well for tensors of order $d=3$, but needs exponentially
increasing storage for $d\rightarrow \infty $. Instead we describe the
hierarchical format. Its storage cost as well as the computational cost of
tensor operations is linear in the dimension $d.$ Hence tensors of size $%
n^{d}$ can be represented (approximated) by $O(dnr+dr^{3}),$ where $r$ is
the bound of the appearing ranks. Using a technique called 'tensorisation',
we can even replace $n$ by $\log n$.
On the closure of the discrete spectrum of compactly perturbed operators
Franz Hanauska
Institut für Mathematik, TU Clausthal, Germany
Abstract:
Let $Z_0$ be a bounded operator on a Banach space $X$ and $K$ a compact operator. Using methods of complex analysis we study the set of accumulation points of the discrete spectrum of the operator $Z:=Z_0+K$. We formulate conditions for $Z$ to exclude certain points or sets to be accumulation points of of the discrete spectrum of $Z$. These results will be applied to compact perturbations of the operator of multiplication and the discrete Laplace operator.
Estimating the number of eigenvalues of linear operators on Banach spaces
Marcel Hansmann
Chemnitz University of Technology, Germany
Abstract:
We will present a new method to obtain upper bounds on the number of eigenvalues of linear operators on Banach spaces. More precisely, we will consider linear operators $L=L_0+K$, which arise from some free operator $L_0$ by a compact perturbation $K$, and derive bounds on the number of eigenvalues of $L$ in the complement of the spectrum of $L_0$.
This talk is based on joint work with M. Demuth, F. Hanauska and G. Katriel.
Approximation of Schrödinger operators with $\delta$-interactions supported on hypersurfaces
Markus Holzmann
TU Graz, Austria
Abstract:
Schrödinger operators with singular $\delta$-interactions supported on hypersurfaces
$\Sigma \subset \mathbb{R}^d$, $d \geq 2$, are used to solve approximately the spectral problem
for classical Schrödinger operators $-\Delta - V$, where the function $V$
has relatively large values in a neighborhood of $\Sigma$ and relatively
small values elsewhere. From a mathematical point of view it is
not clear if these $\delta$-operators, which are formally given by
$A_{\delta, \alpha} := -\Delta - \alpha \delta_{\Sigma}$, have similar spectral properties as classical
Schrödinger operators explained above and how to choose the interaction strength
$\alpha$ to find good approximations.
We are going to answer these questions by constructing a sequence of potentials $V_\varepsilon$
such that the associated Schrödinger
operators $H_\varepsilon := -\Delta - V_\varepsilon$ converge to $A_{\delta, \alpha}$ in the norm
resolvent sense, as $\varepsilon \rightarrow 0$; the connection of the potentials
$V_\varepsilon$ and the interaction strength $\alpha$ of the limit operator can be stated explicitly.
This implies, in particular, that the spectral properties of $A_{\delta, \alpha}$
and $H_\varepsilon$ are approximately the same for sufficiently small $\varepsilon$, which yields
a justification for the usage of $\delta$-operators to solve the spectral problem for
classical Hamiltonians.
This talk is based on a joint work with J. Behrndt, P. Exner and V. Lotoreichik.
Eigenvalues for perturbed periodic Jacobi matrices by the Wigner-von Neumann approach
Edmund Judge
University of Kent, UK
Abstract:
In this talk we discuss a new technique for embedding eigenvalues into the absolutely continuous spectrum of periodic Jacobi operators by adding a potential $(q_n)$. Initially, we make an ansatz for the eigenvector in the form $u_n=Im(\phi_n)\omega_n$, where $\phi_n$ solves the unperturbed problem and the $\omega_n$ are chosen appropriately. We then establish the asymptotic behaviour of the perturbation, $(q_n)$, necessary to realize this assumption. Our calculations show that it is sufficient for $(q_n)$ to be a Wigner-von Neumann potential. The values of the a.c. spectrum for which this method fails are also discussed.
Ramanujan graphs, $\rho$-transience and local tree-likeness
Vadim Kaimanovich
University of Ottawa, Canada
Abstract:
We extend the technique based on considering geodesic spanning trees (earlier applied to Schreier graphs of free groups in a
joint paper with R. Grigorchuk and T. Nagnibeda-Smirnova) to an analysis of Ramanujan graphs and, in particular, show that transience of a Ramanujan graph at the bottom of the spectrum implies that it "looks like a tree" along almost every random path. Further ramifications and conjectures will be discussed. The talk is based on joint work with Tatiana Nagnibeda-Smirnova.
Heinz inequality for the unit ball
David Kalaj
University of Montenegro, Montenegro
Abstract:
In this note we establish the Heinz inequality for harmonic mappings on the boundary of the unit ball by providing a sharp constant $C_n$ in the inequality: $\|\partial_r u(r\eta)\|_{r=1}\ge C_n$, $\|\eta\|=1$, for every proper harmonic mapping of the unit ball onto itself satisfying the condition $u(0)=0$.
Mapping onto poligonal domains with countable set of vertices. Univalence.
Karabasheva Enzhe and Shabalin Pavel
Kazan State University of Architecture and Engineering, Russia
Abstract:
We consider conformal mapping from a half-plane onto a polygonal domain with a countable set of vertices. Preimages of vertices are given. Angles at the vertices of the polygonal domain are known but such vertices are unknown. We apply a solution of the Hilbert boundary value problem with discontinuous coefficients and curling at the infinity of logarithmic order to construct the formula of such conformal mapping. So we have generalization of the Schwarz-Cristoffel integral for this situation.
We search for univalence of such mapping and prove a sufficient condition of univalence.
Opening up and control of spectral gaps for periodic quantum graphs
Andrii Khrabustovskyi
Karlsruhe Institute of Technology, Germany
Abstract:
The name "quantum graph" is used for a pair $(\Gamma,\mathcal{H})$, where $\Gamma$ is a network-shaped structure of vertices connected by edges ("metric graph") and $\mathcal{H}$ is a second order self-adjoint differential operator ("Hamiltonian") on it, which is determined by differential operations on the edges and certain interface conditions at the vertices. Quantum graphs arise naturally in mathematics, physics, chemistry and engineering as models of wave propagation in quasi-one-dimensional systems looking like a narrow neighbourhood of a graph. We refer to the recent book [BK13] containing a comprehensive bibliography on this topic.
In many applications (for example, to graphen and carbon nano-structures) periodic quantum graphs are studied. It is well-known (see, e.g., [BK13, Chapter 4]) that the spectrum of the corresponding Hamiltonians has a band structure, i.e. it is a locally finite union of compact intervals called \textit{bands}. In general the neighbouring bands may overlap. A bounded open interval is called a gap if it has an empty intersection with the spectrum, but its edges belong to it. In general the presence of gaps in the spectrum is not guaranteed. Existence of spectral gaps is important because of various applications, for example in physics of photonic crystals.
The current research concerns spectral properties of a class of periodic quantum graphs. The main peculiarity of the graphs under investigation is that their spectral gaps can be nicely controlled via a suitable choice of the graph geometry and of coupling constants involved in interface conditions at its vertices.
Our main result is as follows [BaKh14].
Theorem. For arbitrary finite intervals $(a_j,b_j)\subset[0,\infty)$ ($j=1,\dots,m$) whose closures are pairwise disjoint and for arbitrary $n\in\mathbb{N}$ we construct a family of $\mathbb{Z}^n$-periodic quantum graphs $\{(\Gamma,\mathcal{H}_\varepsilon)\}_{\varepsilon>0}$ such that the spectrum of $\mathcal{H}_\varepsilon$ has at least $m$ gaps when $\varepsilon$ is small enough, moreover the first $m$ gaps tend to the intervals $(a_j,b_j)$ as $\varepsilon\to 0$.
This is a joint work with Diana Barseghyan (Nuclear Physics Institute, Rež near Prague & University of Ostrava, Czech Republic)
[BaKh14] D. Barseghyan, A. Khrabustovskyi, Gaps in the spectrum of a periodic quantum graph with periodically distributed $\delta'$-type interactions, arXiv:1502.04664 (2014).
[BK13] G. Berkolaiko, P. Kuchment, Introduction to quantum graphs, American Mathematical Society, Providence, RI, 2013.
Stability of crystals
Alexander Komech and Elena Kopylova
Faculty of Mathematics of Vienna University and IITP RAS, Austria
Abstract:
We consider the Schrödinger-Poisson-Newton equations as a model of crystals. Our main results are the well posedness and dispersion decay for the linearized dynamics at the ground state. This linearization is a Hamilton system with nonselfadjoint (and even nonsymmetric) generator. We diagonalize this Hamilton generator using our theory of
spectral resolution of the Hamilton operators with positive definite energy which is a special version of the M. Krein - H. Langer theory of selfadjoint operators in the Hilbert spaces with indefinite metric. Using this spectral resolution, we establish the well posedness and the dispersion decay of the linearized dynamics with positive energy.
The key result of present paper is the energy positivity for the linearized dynamics with small elementary charge $e>0$ under a novel Wiener-type condition on the ions positions and their charge densitities. We give examples of the crystals satisfying this condition.
The main difficulty in the proof of the positivity is due to the fact that for $e=0$ the minimal spectral point $E_0=0$ is an eigenvalue of infinite multiplicity for the energy operator. To prove the positivity we study the asymptotics of the ground state as $e\to 0$ and show that
the zero eigenvalue $E_0=0$ bifurcates into $E_e\sim e^2$.
Eigenvalues asymptotics in a weakly bent leaky quantum wire
Sylwia Kondej
University of Zielona Gora, Institute of Physics, Poland
Abstract:
The main question discussed in the presentation concerns the weak-coupling
behavior of the geometrically induced bound states of singular Schroedinger operators
with an attractive $\delta $ interaction supported by a planar, asymptotically straight curve
$\Gamma $. We show that if $\Gamma $ is slightly bent or weakly deformed
then there is a unique eigenvalue and the gap between it and the continuum threshold is in the
leading order proportional to the fourth power of the bending angle, or the deformation
parameter. Additionally, we analyze the behavior of a general geometrical induced
eigenvalue in the situation when one of the curve asymptotes is wiggled.
The results presented in the talk are based on the common work with P. Exner.
Spectral asymptotics for $2\times 2$ canonical systems
Aleksey Kostenko
University of Vienna, Austria
Abstract:
Based on continuity properties of the de Branges correspondence, we develop a new approach to study the high-energy behavior of $m$-functions and spectral functions of $2\times 2$ first order canonical systems. Our main objective is to provide one-term asymptotic formulas for $m$-functions, as well as spectral functions of canonical systems. In particular, we characterize Hamiltonians such that the corresponding $m$-functions behave asymptotically at infinity as
$$
m_\nu(z)=C{\rm e}^{{\rm i}\pi \frac{1-\nu}{2}}z^{\nu},\quad z\in \mathbb{C}_+,
$$
with some constants $C>0$ and $\nu \in (-1,1)$. Furthermore, we apply these results to radial Dirac and radial Schrödinger
operators as well as to Krein strings and generalized indefinite strings.
The talk is based on a joint work with J. Eckhardt and G. Teschl.
Simultaneously self-adjoint sets of $3\times 3$ matrices
Tomaž Košir
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
Abstract:
For a generic set $\mathcal{M}$ of $3\times 3$ matrices over $\mathbb{C}$ we find necessary and sufficient conditions when $\mathcal{M}$ is simultaneously self-adjoint.
Moreover, for a set of complex hermitean matrices we can tell if there exists a linear combination of matrices which is positive definite. A regular $\mathcal{M}$ can be identified with a determinantal representation of a cubic hypersurface. This allows the use of tools of algebraic geometry. The question of definiteness can be solved using semidefinite programming. We discuss singular case separately.
This is joint work with Anita Buckley.
On the visibility of quantum graph spectra
Christian Kühn
TU Graz, Austria
Abstract:
The spectrum of the Kirchhoff Laplacian on a compact metric graph can be recovered from a corresponding Titchmarsh-Weyl function if the graph satisfies certain geometric conditions. In general this is not true. We will show that nevertheless all eigenvalues below a certain bound can be recovered from the Titchmarsh-Weyl function. This bound can be calculated explicitly from geometric properties of the graph.
This is a joint work with Jonathan Rohleder.
Variational principles for operator functions
Matthias Langer
Department of Mathematics and Statistics, University of Strathclyde, UK
Abstract:
Let $T$ be an operator function which is defined on a complex
domain $\Omega$ and whose values are closed operators in a
Hilbert space. A number $\lambda$ is called an eigenvalue of
the operator function $T$ if there exists a non-zero
$x\in{\rm dom}(T(\lambda))$ such that $T(\lambda)x=0$.
In this talk I will review various variational principles
for eigenvalues (and other parts of the spectrum) of
certain operator functions. Such variational principles can be used
to obtain estimates for eigenvalues or to compare eigenvalues
of two operator functions.
Greedy control
Martin Lazar
University of Dubrovnik, Croatia
Abstract:
Greedy control represents a new notion in the control theory, applicable to parameter dependent systems. It is based on adaptation of (weak) greedy algorithms, developed and explored so far for finding an approximative solutions to parametric PDEs. The idea is to identify the most distinguished parameter values describing the whole range of admissible controls. The algorithm consists of (a possible expensive) offline part devoted to the selection of parameter representatives and the online one enabling a fast computation of an approximative control for a given value of the parameter within a given accuracy.
On spectrum of magnetic graphs
Stepan Manko
Czech Technical University in Prague, Czech Republic
Abstract:
We analyze spectral properties of a quantum graph with a $\delta$ coupling in the vertices exposed to a homogeneous magnetic field perpendicular to the graph plane. We find the band spectrum in the case when the chain exhibits a translational symmetry and study the discrete spectrum in the gaps resulting from compactly supported coupling, magnetic of geometric perturbations. The method we employ is based on translation the spectral problem for the differential equation in question into suitable difference equations. This is a joint project with Pavel Exner.
Optimal normal projections in Krein spaces
Francisco Martínez Pería
Instituto Argentino de Matemática (CONICET) and Depto. de Matemática, Fac. de Cs. Exactas, Universidad Nacional de La Plata, Argentina
Abstract:
In the previous Najman Conference we introduced the $J$-normal projections acting on a Krein space $\mathcal{K}$, i.e. those (bounded) projections $Q$ in $\mathcal{K}$ that commutes with its $J$-adjoint $Q^\#$. A closed subspace $\mathcal{S}$ of $\mathcal{K}$ is the range of a $J$-normal projection if and only if $\mathcal{S}$ is pseudo-regular. Moreover, if the isotopropic part $\mathcal{S}^\circ$ is non-trivial, there exist infinitely many $J$-normal projections onto $\mathcal{S}$ which can be parametrized according to a suitable decomposition of $\mathcal{K}$, see [MMP13].
Along this talk, for a fixed pseudo-regular subspace $\mathcal{S}$ of $\mathcal{K}$, a $J$-normal projection $Q_0$ onto $\mathcal{S}$ is distinguished. Its operator norm can be calculated in terms of the Friedrichs angle between $\mathcal{S}$ and $\mathcal{S}^{[\bot]}$ (in fact, $\|Q_0\|$ is the cosecant of this angle), and it is minimal among the norms of the $J$-normal projections onto $\mathcal{S}$. However, $Q_0$ is not the only $J$-normal projection onto $\mathcal{S}$ with minimal operator norm.
In case that $\mathcal{K}$ is a Minkowski space (i.e. $\dim(\mathcal{K})<\infty$), $Q_0$ is the only $J$-normal projection that minimizes every unitarily invariant norm defined on $\mathcal{K}$. This property has a significant geometrical interpretation: given any other $J$-normal projection $Q$ onto $\mathcal{S}$, the principal angles between $\mathcal{S}$ and $\ker(Q_0)^\bot$ are (one-to-one) bigger or equal than the corresponding principal angles between $\mathcal{S}$ and $\ker(Q)^\bot$.
This talk is based on a joint work with J. I. Giribet and A. Maestripieri [GMMP15].
References:
[G] A. Gheondea, On the geometry of pseudo-regular subspaces of a Krein space, Op. Theory: Adv. and Appl. bf 14 (1984), 141-156.
[MMP13] A. Maestripieri and F. Martínez Pería; Normal projections in Krein spaces, IEOT 76 (2013) 357-380.
[GMMP15] J.I. Giribet, A. Maestripieri, and F. Martínez Pería; Optimal normal projections in Krein spaces, preprint.
Existence of strong traces for entropy solutions to degenerate parabolic equations
Darko Mitrović
University of Zagreb, Croatia
Abstract:
We prove that an entropy solution $u$ to a degenerate parabolic equation admits strong trace $u_0$ at $t=0$ in the sense that $u(t,\cdot)$ strongly converges in $L^1_{loc}(R^d)$ toward $u_0$ as $t\to 0$. The basic tool is a variant of the H-measures.
Block diagonalization of $J$-self-adjoint operators and resonances
Alexander K. Motovilov
Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia
Abstract:
We study the block operator matrices 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 on 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 spectrum of $A_0$ overlaps the absolutely
continuous spectrum of $A_1$. We formulate conditions ensuring the
existences of a bounded solution $X:\mathfrak{H}_0\to\mathfrak{H}_1$
to the associated operator Riccati equation
$$
XA_0-A_1X+XBX=-B^*
$$
and then the block diagonalization of $L$ in terms of the
corresponding invariant subspaces that are the graphs of $X$ and
$X^*$. The above conditions involve the absence of resonances for
the Schur complement $M(z)=A_0-z+B(A_1-z)^{-1}B^*$,
$z\in\mathbb{C}\setminus\mathop{\rm spec}(A_1)$, in certain domains
adjacent to the absolutely continuous spectrum of $A_1$ in the
so-called unphysical sheets of the spectral parameter plane.
Our approach to the problem is based on the ideas and results
worked out in [1-3].
[1] S. Albeverio, K. A. Makarov, and A. K. Motovilov:
Graph subspaces and the spectral shift
function, Canad. J. Math, 55 (2003),
449-503.
[2] S. Albeverio and A. K. Motovilov:
Operator Stieltjes integrals with respect to a spectral
measure and solutions of some operator equations,
Trans. Moscow Math. Soc. 72
(2011), 45-77.
[3]
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.
On the pseudospectrum of the harmonic oscillator with imaginary cubic potential
Radek Novak
Nuclear Physics Institute, Academy of Sciences of the Czech Republic
Abstract:
We study the Schrödinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and its spectrum is provided and the importance of examining its pseudospectrum as well is emphasized. This is achieved by employing scaling techniques and treating the operator using semiclassical methods. The existence of pseudoeigenvalues very far from the spectrum is proven, and as a consequence, the spectrum of the operator is unstable with respect to small perturbations and the operator cannot be similar to a self-adjoint operator via a bounded and boundedly invertible transformation. It is shown that its eigenfunctions form a complete set in the Hilbert space of square-integrable functions; however, they do not form a Schauder basis.
Bonsall's spectral radius and the approximate point spectrum
Aljoša Peperko
Faculty of Mechanical Engeenering, University of Ljubljana, Slovenia
Abstract:
The corollary of our main result states that the Bonsall's cone spectral radius of a (in general non-linear) positively homogeneous, bounded and supremum preserving map, defined on a max-cone in a given Banach lattice, is always included in its approximate point spectrum. This result applies to a large class of (nonlinear) max-type operators (in the infinite and finite dimensional setting) appearing in problems arising in certain differential and difference equations, combinatorial optimisation, mathematical physics, mathematical biology (DNA analysis), scheduling, parallel computing, ...
By applying the techniques of a proof of the above result, we are also able to prove a generalization of a an (implicitly) known result that the spectral radius of a positive (linear) operator on a Banach lattice is contained in the approximate point spectrum.
Joint work with Vladimir Müller.
On Abstract $\mathbf{grad}$-$\mathbf{div}$ Systems
Rainer Picard
TU Dresden, Dresden, Germany
Abstract:
In the framework of a particular class of dynamical systems based
on the investigation of typical systems of partial differential equations,
skew-selfadjointness of the spatial operator $A$ involved plays an
important role as a key reference case for modeling physical phenomena,
see e.g. [1]. Typically, skew-selfadjointness hinges
on the particular form of $A$ as
$$
A=\left(\begin{array}{cc}
0 & -G^{*}\\
G & 0
\end{array}\right),
$$
where $G:D\left(G\right)\subseteq H_{0}\to H_{1}$ is a closed densely
defined linear operator between the Hilbert spaces $H_{0},\:H_{1}$.
A typical case is where $G=\mathrm{grad}=\left(\begin{array}{c}
\partial_{1}\\
\partial_{2}\\
\partial_{3}
\end{array}\right)$ realized as the operator of weak differentiation from $L^{2}\left(\Omega\right)$
to $L^{2}\left(\Omega\right)^{3}$, where $\Omega$ is a non-empty
open set in 3-dimensional Euclidean space. In this case $-G^{*}$
is the weak divergence operator with containment in $D\left(G^{*}\right)$
generating a generalization of vector fields being ``tangential''
even though a normal may not exist. The particular class of operators
$A$ under consideration is an abstraction of this situation, hence
the term ``abstract $\mathrm{grad}$-$\mathrm{div}$ systems'',
where the role of the partial derivatives $\partial_{k},\:k=1,2,3$,
is replaced by general Hilbert space operators. The utility of the
approach is demonstrated by examples.
[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.
Numerical methods for separable boundary value problems using spectral collocation and multiparameter eigenvalue problems
Bor Plestenjak
University of Ljubljana, Slovenia
Abstract:
In numerous science and engineering applications a partial differential equation has to be solved on a domain that allows the use of the method of separation of variables. In several coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP), some important cases are Mathieu’s system, Lamé’s system, and a system of spheroidal wave functions. Up to now, multiparameter approaches have rarely been exploited to solve the equations numerically. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs, where we exploit the structure of the matrices, it is possible to solve such problems both very efficiently and accurately. Related numerical methods are available in MultiParEig - toolbox for multiparameter eigenvalue problems in Matlab.
Operator spectrum and biophysical systems in external fields
Igor Yu. Popov
ITMO University, St. Petersburg, Russia
Abstract:
Solvable models based on the theory of self-adjoint extensions of symmetric operators are suggested for two biophysical systems. First, we consider the lateral line organ of a fish. It consists of a set of knobs (neuromasts) disposed along some lines on a head and trunk of a fish. We suggest a model allowing one to describe the process of sound detection by this sensitive organ. In the model, we deal with two parallel chains of acoustic resonators. The Robin boundary condition is assumed The spectrum of the corresponding Laplacian is analyzed. It is shown that for some model parameters, there exists a band close to the origin. Due to this spectral property, one can observe resonance phenomenon for physically reasonable range of acoustic frequencies. Moreover, the resonance allows one to determine the direction of the incoming wave propagation. Due to the existence of two lateral lines (two chains of resonators) one has, in principle, an opportunity to find the distance from the sound source. The comparison of two types of lateral line organs from the point of view of sound detection is made.
The second considered biophysical system is DNA molecule. We deal with a quantum graph having the form of double helix connected by bridges (segments) in a magnetic field. The spectrum of the electron Hamiltonian is studied. Its dependence on the magnetic field is described.
Spectral analysis of generalized linear viscoelasticity models
Victor Vlasov and Nadezhda Rautian
Lomonosov Moscow State University, Russia
Abstract:
We study integro-differential equations with unbounded operator coefficients in Hilbert space. The principal part of these equations is an abstract hyperbolic operator perturbed by summands of Volterra integral operators. Operator models of such type have many applications in the linear viscoelasticity theory, homogenezation theory, heat conduction theory in media with memory, etc. In particular these integro-differential equations can be realized as the system of integro-partial differential equations:
$$
\rho \ddot u(x,t) - Lu(x,t) + \int\limits_0^t {{\Gamma _1}(t - s)L_1u(x,s)ds} + \int\limits_0^t {{\Gamma _2}(t - s)L_2u(x,s) u(x,s)ds} = f(x,t),
$$
were $u = \vec u(x,t) \in {\mathbb{R}^3}$ is displacement vector of viscoelastic anisotropic media, $t>0$, $x \in \Omega \subset {\mathbb{R}^3}$ is bounded domain, $u$ satisfy Dirichlet conditions in a domain with smooth boundary $\Omega $, $Lu = (L_1+L_2)u=\mu \cdot (\Delta u + {\text{grad div}}u) +\lambda \cdot {\text{grad div}}u$ is Lame operator of elasticity theory, ${\Gamma _1}$, ${\Gamma _2}$ are memory relaxation functions that are the series of decreasing exponents with positive coefficients.
A spectral analysis of the operator-valued functions, which are the symbols of considered integro-differential equations is provided. The structure and localization of spectra for these operator-valued functions are analyzed.
These results are the natural generalization of our results obtained in [1],[2].
References:
[1] V. V. Vlasov, N. A. Rautian and A. S. Shamaev Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics, Journal of Mathematical Sciences
190:1 (2013) 34-65.
[2] Vlasov V. V., Rautian N. A. Spectral Analysis and Representations of Solutions of Abstract Integro-differential Equations in Hilbert Space, Operator Theory: Advances and Applications 236 (2013) 519-537, Springer Basel AG.
On maximal parabolic regularity
Joachim Rehberg
Weierstrass-Institut Berlin, Germany
Abstract:
In recent years maximal parabolic regularity turned out to be an extremely powerful tool for the treatment of linear and nonlinear parabolic equations — in particular when the time-dependency is nonsmooth. We present scales of functional spaces where second-order divergence operators do satisfy maximal parabolic regularity, even if the domain is non- smooth, the coefficients are discontinuous and the boundary conditions are mixed. Finally, we indicate applications of this for nonlinear problems, according to Pruess’ theorem.
Recovering quantum graph spectrum from vertex data
Jonathan Rohleder
Graz University of Technology, Austria
Abstract:
We discuss the question to what extent spectral information of a Schrödinger operator on a finite, compact metric graph subject to standard or $\delta$-type matching conditions can be recovered from a corresponding Titchmarsh-Weyl function on the boundary of the graph. In contrast to the case of ordinary or partial differential operators, the knowledge of the Titchmarsh-Weyl function is in general not sufficient for recovering the complete spectrum of the operator (or the potentials on the edges). However, we show that those eigenvalues with sufficiently high (depending on the cyclomatic number of the graph) multiplicities can be recovered. Moreover, we prove that under certain additional conditions the Titchmarsh-Weyl function contains even the full spectral information.
J. Rohleder, Recovering a quantum graph spectrum from vertex data, J. Phys. A: Math. Theor. 48 (2015) 165202 (20pp).
A quantitative Carleman estimate for elliptic second order partial differential operators
Christian Rose
Technische Universität Chemnitz, Germany
Abstract:
We present a Carleman inequality of the form
$$
\int \alpha w^{1-2\alpha}\vert\nabla u\vert^2+\alpha^3 w^{-1-2\alpha}\vert u\vert^2\leq C\int w^{2-2\alpha}\vert Lu\vert^2,
$$
for suitable $\alpha\geq \alpha_0>0$ and $C>0$, where we integrate over a punctured ball $B$ of radius one, $u\in\mathcal{C}_c^\infty(B)$, $L$ is an elliptic second order partial differential operator with Lipschitz coefficients and $w$ a suitable weight function. The usefulness of our estimate is given by the explicit dependence of the parameters $\alpha_0$ and $C$ and the weight function $w$ in terms of the ellipticity and Lipschitz constants of the involved operator $L$. More generally, our result can be stated as a scaled version depending on the radius of the ball and which is valid for complex-valued $W^{2,2}$-functions. Additionally, it is possible to treat operators with bounded first and zeroth order terms. Such estimates lead to quantitative unique continuation principles for elliptic second order operators with slowly varying coefficients.
This is joint work with I. Nakić and M. Tautenhahn.
Multiplicative Lidskii's inequalities and its applications to frame theory
Mariano Ruiz
Universidad Nacional de La Plata & Instituto Argentino de Matemática, Argentina
Abstract:
In this talk, we consider a multiplicative analogue of Lidskii's inequality in terms of log-majorization and a characterization of the case of equality. These results are used to study two design problems in frame theory:
on the one hand, given a fixed finite frame $\mathcal{F}=\{f_j\}_{j=1}^n$ for $\mathbb{C}^d$, to compute those dual frames $\mathcal{G}$ of $\mathcal{F}$ that
are optimal perturbations of the canonical dual frame for $\mathcal{F}$ under certain restrictions on the norms of the elements of $\mathcal{G}$.
On the other hand,
the problem of characterize those $V\cdot \mathcal{F}=\{V\,f_j\}_{j=1}^n$ - for
invertible operators $V$ which are close to the identity - that are optimal perturbations of $\mathcal{F}$. That is, to compute the optimal perturbations of $\mathcal{F}$ among frames $\mathcal{G}=\{g_j\}_{j=1}^n$ that have the same linear relations as $\mathcal{F}$.
In both cases, optimality is measured with respect to submajorization of the eigenvalues of the frame operators. Hence, our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error.
This contribution is joint work with Pedro Massey (UNLP-IAM, Argentina) and Demetrio Stojanoff (UNLP-IAM, Argentina).
Dynamical and spectral Dirac systems: response and Weyl functions, inverse problems
Alexander Sakhnovich
University of Vienna, Austria
Abstract:
We consider interconnections between response and Weyl functions (and A-amplitudes) for Dirac systems.
As a result we give explicit and general type solutions of inverse problem to recover system from response
function. The results are based on the recent paper [1] and our book [2].
[1] M.I. Belishev and V.S. Mikhailov, Inverse problem for a one-dimensional dynamical Dirac system (BC-method). Inverse Problems 30 (2014), 125013, 26 pp.
[2] A.L. Sakhnovich, L.A. Sakhnovich, and I.Ya. Roitberg, Inverse problems and nonlinear evolution equations.
Solutions, Darboux matrices and Weyl-Titchmarsh functions. De Gruyter Studies in Mathematics 47, De Gruyter, Berlin, 2013.
The Anderson Model on the Bethe Lattice: Lifshitz Tails
Christoph Schumacher
TU Chemnitz, Germany
Abstract:
The Anderson model is the prototypical random Schroedinger operator and used to model crystals with impurities and alloys.
Mathematically, the Anderson Hamiltonian is the sum of the Laplace operator and a random potential.
In this talk, we deal with the discrete Anderson operator on the Bethe lattice, which is the infinite tree graph with constant degree.
Our object of interest is the behaviour of the Integrated Density of States (IDS) of the Anderson Hamiltonian at the bottom of the spectrum.
More specifically, we prove Lifshitz tail behavior, i.e. doubly exponential decay with exponent 1/2.
Due to the exponential volume growth of the balls, the spectral theory of the free Laplace operator on the Bethe lattice fundamentally differs from Euclidian lattices.
Therefore, the elegant proof of Lifshitz tails in Euclidian lattices fails on the Bethe lattice.
Instead, we study the Laplace transform of the IDS, employ Tauberian theorems, a discrete Feynman-Kac formula, a discrete IMS localization formula, spectral theory of finite symmetric rooted trees, an uncertainty principle for low-energy states, and epsilon-net argument and concentration inequalities.
In the talk, I will present part of the proof.
The full proof can be found here.
This is joint work with Francisco Hoecker-Escuti.
Principal solutions at infinity for nonoscillatory linear Hamiltonian systems
Roman Simon Hilscher
Masaryk University, Czech Republic
Abstract:
This is a joint work with Peter Šepitka. In this talk we will discuss the theory of principal solutions at infinity for nonoscillatory linear Hamiltonian systems. These are in a certain sense the smallest solutions of the systems at infinity. A key new ingredient is that we do not assume the complete controllability (or identical normality) of the system. We show how to define the principal solutions at infinity for this more general case and that the principal solutions can have their rank equal to any integer value in an explicitly given range. The smallest rank corresponds to the unique minimal principal solution at infinity, while the largest rank corresponds to the traditional maximal (i.e. invertible) principal solution at infinity. We shall comment on some applications of the principal solutions at infinity for controllable systems in the oscillation and spectral theory and seek for such applications in the abnormal case.
Transition in the Microscopic Eigenvalue Distribution of Various Classes of Operators
Mihai Stoiciu
Williams College, USA
Abstract:
We investigate the microscopic distribution of the eigenvalues for various classes of self-adjoint and unitary operators: discrete Schrödinger operators, Jacobi matrices, and their unitary analog, the CMV matrices. We show that random CMV matrices exhibit a transition in the microscopic eigenvalue distribution from Poisson to the equidistant "clock" distribution, via the circular beta ensemble of the random matrix theory. Deterministic CMV operators corresponding to Patterson-Sullivan measures associated to hyperbolic reflection groups exhibit a similar transition towards the clock eigenvalue distribution. Finally, we consider one-dimensional random Schrödinger operators with small coupling constants and we describe the behavior of the density of states and the transition in the microscopic eigenvalue distribution of these operators, as the coupling constant approaches 0. These results were obtained in joint projects with Peter Hislop, Rowan Killip, and Norbert Peyerimhoff.
Coherency preservers
Peter Šemrl
University of Ljubljana, Slovenia
Abstract:
We will present a description of continuous coherency preservers on hermitian matrices. Applications in mathematical physics will be discussed.
Recessive solutions for nonoscillatory discrete symplectic systems
Peter Šepitka
Masaryk University, Brno, Czech Republic
Abstract:
This is a joint work with Roman Šimon Hilscher. In this talk we present
a new concept of a recessive solution for discrete symplectic systems without
any eventual controllability assumption. We show that the existence of a recessive
solution is equivalent with the nonoscillation of the system and that recessive
solutions can have any rank between explicitly given lower and upper bounds.
The smallest rank is associated with the minimal recessive solution, which is
unique up to a right nonsingular multiple. The largest rank leads to the traditional
maximal recessive solution. Presented results are new even for special discrete
symplectic systems, such as for even order Sturm–Liouville difference equations
and linear Hamiltonian difference systems.
On a product of distributions with applications
Jela Šušić
University of Montenegro, Montenegro
Abstract:
We introduce a way to multiply two $\delta$-distributions and apply it to prove existence of a physically reasonable solution to a photon distribution equation.
Eigenvalue problem for one-dimensional model of elastic stents
Josip Tambača
Department of Mathematics, University of Zagreb, Croatia
Abstract:
In this work we formulate the eigenvalue problem for a linearized one-dimensional model of elastic stents.
Since stents are mostly metallic structures their mechanical behavior is modeled, using the elasticity theory,
as a union of three-dimensional struts. If the deformations in the problem are small, the behavior of stents can
be modeled by the three--dimensional linearized elasticity. These equations of linearized elasticity in thin domains
are very demanding for numerical approximation and qualitative analysis. More simple analytical approximation
can be made using one-dimensional model of curved elastic rods for thin struts.
Additionally, we need to prescribe the coupling conditions that need to be satisfied at each vertex of the stent net
where the edges (stent struts) meet.
There are two sets of coupling conditions:
- the kinematic coupling condition: displacement and infinitesimal rotations are continuous at each vertex,
- the dynamic coupling condition: balance of contact forces and contact moments at each vertex.
Proposed model for thin struts (edges) and the contact condition fully defines the stent model.
Thus, for the whole structure we obtain a system of ordinary differential equations on a graph.
Associated eigenvalue problem is then discretized using mixed finite element method and the associated discrete approximation is solved.
Obtained eigenvalues and eigenvectors are then compared for four different coronary stents commercially available at the market
and certain mechanical properties are deduced.
This is a joint work with Luka Grubišić and Josip Iveković.
Quantitative unique continuation for Schrödinger operators and applications
Martin Tautenhahn
Technische Universität Chemnitz, Germany
Abstract:
In this talk we present a scale free and quantitative unique continuation principle for linear combinations of eigenfunctions of Schrödinger operators.
Let $\Lambda_L = (-L,L)^d$ and $H_L = -\Delta + V$ be a Schrödinger operator on $L^2 (\Lambda_L)$ with a bounded potential $V : \mathbb{R}^d \to [-K,K]$ and Dirichlet, Neumann or periodic boundary conditions. Our main result is of the type
$$
\lVert \phi \rVert_{\Lambda_L}^2 = C_{\rm sfuc} \lVert \phi \rVert_{W \cap \Lambda_L}^2,
$$
where $\phi = \sum_{E_k \in (-\infty,b]} \alpha_k \phi_k$ is a complex linear combination of eigenfunctions corresponding to eigenvalues in $E_k \in (-\infty,b]$, $W$ is some equidistributed set in $\mathbb{R}^d$, and $C_{\rm sfuc}$ is a constant depending only on $d$, $b$, $K$ and some property of $W$. In particular, the constant $C_{\rm sfuc}$ is independent of $L$ and the dependence on the other parameters is known explicitly.
We briefly discuss two applications of this result. First, we establish scale-free controllability for the heat equation, and second, we obtain a Wegner estimates for a class of random operators called the random breather model.
Conditional Wegner Estimate for the Random Breather Model
Matthias Täufer
TU Chemnitz, Germany
Abstract:
We prove a conditional Wegner estimate for Schrödinger operators with constant magnetic field and random potentials of breather type. More precisely, the proof of the Wegner estimate is reduced to a scale free unique continuation principle. The relevance of such unique continuation principles has been emphasized in the last years and in the case of vanishing magnetic field, such an estimate has recently been proven by Nakić, T., Tautenhahn and Veselić.
The breather type potential is a prototypical example of a random Schrödinger operator where the randomness enters in a non-linear, but still monotonous fashion.
We focus on the standard breather model, meaning that the single site potential is the characteristic function of a ball or a cube. However, our method applies to a substantially larger class of radially decaying single-site potentials which we comment on.
Bounds for the Spectrum of Non-Selfadjoint Block Operator Matrices and Applications
Axel Torshage
Department of Mathematics and Mathematical Statistics, Umeå universitet, Sweden
Abstract:
In this talk we present spectral and perturbation theory of a class of selfadjoint block operator matrices in a Krein space. The perturbation is a bounded non-symmetric operator and we classify the spectrum of the perturbed operator and establish bounds on the complex eigenvalues. Furthermore, we show that these bounds are optimal and analyze the behavior of the bounds in detail. From the bounds, it is deduced that the imaginary part approaches zero in at least quadratic pace as the real part of the eigenvalues approaches infinity. Moreover, we apply the new theory to an unbounded rational operator function with origin in electromagnetic field theory. The analysis applies for example to electromagnetic wave propagation in metallic photonic crystals and microwave cavities containing lossy dielectric materials.
This is a joint work with Christian Engström.
Variational principles for eigenvalues in spectral gaps and applications
Christiane Tretter
Mathematisches Institut, Universitaet Bern, Switzerland
Abstract:
In this talk variational principles for eigenvalues in gaps of the essential spectrum are presented which can be used to establish two-sided eigenvalue bounds.
The results are applied to a spectral problem modelling photonic crystals and to the Klein-Gordon equation, in reminiscence of Branko Najman.
Nonlinear boundary conditions and a class of m-accretive block operator matrices
Sascha Trostorff
TU Dresden, Germany
Abstract:
We consider (nonlinear) restrictions of block operator matrices of the form
$$
\begin{pmatrix}
0 & D \\
G & 0
\end{pmatrix},
$$
where $G$ and $D$ are densely defined closed linear operators on Hilbert spaces satisfying $G^\ast \subseteq - D$. We characterize all m-accretive restrictions of those operator matrices in terms of so-called abstract boundary data spaces associated with $G$ and $D$.
On Quadratic Eigenvalue Problem
Ninoslav Truhar
Department of Mathematics, University of Osijek, Osijek, Croatia
Abstract:
We consider the two related problems connected with the quadratic eigenvalue problem (QEP):
$$(\mu^2 M + \mu C + K) x =0 \,. $$
The first problem refers to the relative perturbation bounds for the eigenvalues as well as for the eigensubspaces of the considered QEP,
where $M, C, K \in \mathbb{C}^{n\times n}$ are given Hermitian matrices, such that
all eigenvalues of QEP are real, that is the quadratic eigenvalue problem is so called hyperbolic. Using a proper linearization and the new relative perturbation bounds for obtained definite matrix pairs, we develop a novel relative perturbation bounds which are uniform and depend only on matrices $M$, $C$, $K$ perturbations $\delta M$, $\delta C$ and $\delta K$ and the standard relative gaps.
Within the second problem, we present a novel approach to the problem of Direct Velocity Feedback (DVF) optimization of vibrational structures, which can be written as
$$
M \ddot{x} + v b b^T \dot{x} + K x = 0\,,
$$
where $M, K\in \mathbb{R}^{n\times n}$ are symmetric positive definite matrices,
$b\in \mathbb{R}^{n\times r}$ and $v\in \mathbb{R}$ (usually $v>0$).
The above problem leads to the parametric eigenvalue problem, thus we describe the behavior
of the eigenvalues for a small and large gains separately, that is for a small and a large $v$. For the small gains, which are connected to a modal damping approximation, we present a standard approach based on Gerschgorin discs. For the large gains we present a new approach which allows us to approximate all eigenvalues very accurately and efficiently.
Besides mentioned application the new bounds are also valuable for better understanding of
damped vibration systems where eigenvalue behavior plays important role.
A functional analytic look upon Remling's oracle theorem
Hendrik Vogt
Universität Bremen, Germany
Abstract:
Remling's oracle theorem [1] is a corner stone in the study of
the absolutely continuous spectrum of one-dimensional Jacobi operators
and Schrödinger operators. Roughly speaking, it says that
one-dimensional operators with absolutely continuous spectrum always
are "somewhat predictable".
The proof given by Remling relies on limit properties of the
asymptotic value distribution of the Titchmarch-Weyl $m$-function
belonging to the operator. We indicate a different approach based
on the Poisson transform on the half plane, and we explain how
the method can be applied to general Sturm-Liouville operators.
(joint work with Christian Seifert)
References:
[1]
Christian Remling,
The absolutely continuous spectrum of Jacobi matrices,
Ann. of Math. 174 (2011), no ,1, 125-171.
On the resolvent regularization by Witten
Marcus Waurick
TU Dresden, Germany
Abstract:
For a Fredholm operator $L$ in a Hilbert space $H$, a possible method to compute the index $\textrm{ind}\, L$ is to use the resolvent regularization introduced by Witten:
$$
\textrm{ind}\, L=\lim_{z\to 0+} z\textrm{tr}_H\left(\left(L^*L+z\right)^{-1}-\left(LL^*+z\right)^{-1}\right),
$$
provided $\left(\left(L^*L+z\right)^{-1}-\left(LL^*+z\right)^{-1}\right)$ is a trace class operator in $H$.
We propose a generalization of this formula and apply it to the computation of the index of the Dirac operator on $\mathbb{R}^n$, $n\geq 3$ odd, with some potential added being invertible outside a ball, satisfying suitable decay conditions on its derivatives.
Computations for the index of the Dirac operator with suitable potential, which are available in the literature, either use completely different methods (e.g. K-Theory) or suffer from serious flaws.
This is joint work with Fritz Gesztesy from the University of Missouri, Columbia, USA.
Some spectral results for waveguides
Ian Wood
University of Kent, UK
Abstract:
We study a spectral problem for the Laplacian in a weighted space which is related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We use variational arguments to prove that guided mode spectrum can be created by arbitrarily small perturbations to the coefficient in the equation. After performing a Floquet decomposition in the axial direction of the waveguide, we study the spectrum created by the perturbation for any fixed value of the quasi-momentum.
Rank 2 deformations of operators and relations with non-commutative probability
Janusz Wysoczanski
Institute of Mathematics, Wroclaw University, Poland
Abstract:
We shall present a family of rank 2-deformations of operators, and study the properties of the transformed spectra. Several special cases will be considered, among which one subfamily will be related to the t-deformation (of measures) defined by Bozejko and Wysoczanski in the framework of non-commutative probability, and to its generalization defined by Krystek and Yoshida.. In particular, we shall present the deformations related to the family of free Meixner laws, which are the free probability analogues of the classical Meixner laws.
This talk is based on joint work with Anna Kula (Wroclaw University) and Michal Wojtylak (Jagiellonian University, Krakow).
Dichotomy of Hamiltonians for PDE systems with boundary control and observation
Christian Wyss
University of Wuppertal, Germany
Abstract:
We consider the Hamiltonian operator matrix
$$T=\begin{pmatrix}A&-BB^*\\-C^*C&-A^*\end{pmatrix}$$
from control theory.
Here $A$ is a partial differential operator generating a $C_0$-semigroup
on
a bounded domain $\Omega$, while
the control and observation operators $B$ and $C$ act on the boundary
$\partial\Omega$.
Connected to the Hamiltonian is the algebraic Riccati equation
$$A^*X+XA-XBB^*X+C^*C=0.$$
Its solutions $X$ are in one-to-one correspondence to invariant
graph subspaces of $T$.
We will construct such subspaces using a dichotomy property of $T$:
Under appropriate conditions on $A,B$ and $C$ a strip around the imaginary axis
belongs to the resolvent set of $T$ and there exist invariant subspaces
$V_+$ and $V_-$ corresponding to the spectrum in the right and left
half-plane, respectively.
The proof is based on a new abstract perturbation result for dichotomy,
which accounts for the problem that $BB^*$ and $C^*C$ map into
extrapolation spaces.
We then use certain Krein space symmetries of $T$ to show that $V_+$ and
$V_-$ are graph subspaces and thus yield solutions of the
Riccati equation.