# Talks and posters of Analysis chair members

Showing search results 1–255 of 255.
2022
• Spectral inequalities with sensor sets of decaying density., Oldenburg, September 26–30.

We discuss spectral inequalities for the harmonic oscillator and more general Schrödinger operators with confinement potentials on the whole space. It turns out that the (super-)exponential decay of the corresponding eigenfunctions allows to consider sensor sets with a density that exhibits a certain decay. This, in particular, permits sensors with finite measure. This talk is based on joint work with A. Dicke and I. Veselić.

• Spectral inequalities and observability with sensor sets of decaying density., TU Hamburg-Harburg, July 11.

We discuss spectral inequalities and observability for the harmonic oscillator and more general Schrödinger operators with confinement potentials on the whole space. It turns out that the (super-)exponential decay of the corresponding eigenfunctions allows to consider sensor sets with a density that exhibits a certain decay. This, in particular, permits sensors with finite measure. This talk is based on joint work with A. Dicke and I. Veselić.

• Spektralungleichungen mit Sensormengen abfallender Dichte., Bergische Universität Wuppertal, May 30.
• Spectral inequality for Shubin-type operators., TU Dortmund, May 31.

Decay properties of eigenfunctions of Shubin-type operators, i.e., Schrödinger operators with potentials of the form $|x|^\tau$ with $\tau > 0$, are examined. After that, a spectral inequality for these operators with sensor sets of finite Lebesgue measure is shown and put into context. The presented results are based on joint work with A. Seelmann and I. Veselić.

• Logvinenko-Sereda type theorems for spectral subspaces., TU Dortmund, April 5.

Results on spectral inequalities based on techniques from complex analysis are reviewed. Special emphasis is paid to the pure Laplacian on Euclidean domains and (partial) harmonic oscillators on $\mathbb{R}^d$, recently discussed in joint works with M. Egidi, A. Dicke, and I. Veselić.

• Observability from sensor sets with decaying density., FernUniversität in Hagen, April 6. [URL]

We show observability for a whole class of parabolic equations on $\mathbb{R}^d$ from sensor sets with decaying density. The proof is based on spectral inequalities for the (partial) harmonic oscillator and corresponding dissipation estimates. The presented results are based on joint work with Albrecht Seelmann and Ivan Veselić.

• Observability for the (anisotropic) Hermite semigroup from finite volume or decaying sensor sets., TU Hamburg, 07. February.

This is joint work with A.Dicke and A. Seelmann. We study the observability and null control problem for the semigroup generated by the harmonic oscillator and the partial harmonic oscillator. We identify sensor sets which ensure null controlabillity improving and unifying previous results for such problems. In particular, it is possible to observe the Hermite semigroup from finite volume sensor sets.

• Observability for the Hermite semigroup from finite volume sensor sets., Institut de recherche mathématique de Rennes, 20. January.
2021
• Group testing, Steiner Systems, and Reed-Solomon Codes., Hagen, 2021–02–18. [URL]
• Wegner estimate for random divergence-type operators., FernUniversität in Hagen, November 17.

Random divergence-type operators are second-order elliptic operators where the second-order term is randomly perturbed by some non-negative function. The model we study in this talk includes random perturbations that may depend on the the random parameters in a non-linear way. We discuss the proof of the Wegner estimate for these operators and show that it can be derived from a quantitative unique continuation estimate for the gradient of an eigenfunction of a divergence-type operator. The results are based on joint work with Ivan Veselić.

• On Alexander Logunov's technique., TU Dortmund, May 11.
• Uncertainty relations and applications., Universität Rostock, 15. December.

Uncertainty relations or unique continuation estimates for various classes of functions are investigated in several fields of mathematical analysis. They have also a number of interesting applications, among them those in mathematical physics and the theory of partial differential equations. While in other areas uncertainty implies less knowledge or weaker mathematical results, in these cases uncertainty principles in fact improve our knowledge about certain mathematical objects. The talk aims at shedding a bit of light on these aspects of uncertainty principles.

• Quantitative uncertainty relations in applied analysis and mathematical physics., University of Sussex, 28. October.

We recall several uncertainty relations and present a version for solutions of elliptic PDEs in R^d. It plays a crucial role in the mathematical theory of Anderson localization and in the control theory of the heat equation. In a detour, we discuss also the role of uncertainty relations in compressive sensing.

• Scale free unique continuation estimates and applications for periodic and random operators., Texas A&M, 6. October.

With Ivica Nakic, Matthias Taeufer and Martin Tautenhahn we established a quantitative unique continuation estimate for spectral projectors of Schroedinger operators. It compares the L^2 norm of a function in a spectral subspace associated to a bounded energy interval to the L^2 norm on an equidistributed set. These estimates allow to give quantitative two-sided bounds on the lifting of edges of bands of essential spectrum, as well as on discrete eigenvalues between two such bands. It also allows to deduce Anderson localization in regimes where this was not possible before. For instance, Albrecht Seelmann and Matthias Taeufer showed that Anderson localization occurs at random perturbations of band edges of periodic potentials, whether the edges exhibit regular Floquet eigenvalue minima or not.

• Uncertainty principles and lifting of eigenvalues., Universität Leipzig, 14. June, 15:00.

We present two results which fit together nicely. The first belongs to the realm of partial differential equations and quantifies "propagation of smallness" or "unique continuation" for linear combinations of eigenfunctions of Schroedinger operators. The second is an operator theoretic result, describing the lifting of eigenvalues and edges of bands of essential spectrum. The combination of these two results yields two-sided Lipschitz bounds on the movement of spectral edges and discrete eigenvalues, both below and in gaps of the esential spectrum.

• Null controllability for the semigroup of the harmonic oscillator., Dubrovnik, Croatia, 14. June , 09:00.

More than 15 years ago it was established that control on any set of positive Lebesgue measure is sucient to drive the solution of the free heat equation on a bounded domain with Dirichlet boundary conditions to zero at any prescribed positive time. More recently, it was established that the free heat equation in the whole euclidean space is null controllable iff the sensor set is thick. In the first case the spatial domain is bounded and the generator has purely discrete spectrum. In the second case the spatial domain is unbounded and the generator has purely continuous spectrum. Our interest is to reconcile and interpolate these two phenomena. This can be done, on one hand, by a quantitative analysis of the control cost estimates and their dependence on the geometric features of the spatial domain and the sensor set. On the other hand, the control problem for the semigroup generated by the harmonic oscillator exhibits a mix of the phenomena spelled out above: The spatial domain is unbounded, but the generator has purely discrete spectrum. We present new uncertainty principles of Hermite functions that imply null controllability for sensor sets that are much sparser than thick sets. This is joint work with A. Dicke and A. Seelmann.

• Wegner estimates for Gaussian random potentials (under minimal assumptions)., TU Chemnitz, Faculty of Mathematics, 12. May , 15:00 CET.
• Three applications of scale uniform uncertainty relations for Schrodinger equations., Kassel, 18. March , 08:30–09:10.

I will present recent results on quantitative unique continuation estimates for functions in appropriately chosen subspaces which lead to uncertainty relations and so-called spectral inequalities, respectively. The mentioned appropriately chosen subspaces could be defined in terms of the properties of the Fourier transform or as spectral subspaces of an elliptic second order operator. The problems which I consider are defined on the whole Euclidean domain or on large boxes, which may be considered as an approximation of the whole space. The obtained estimates are uniform over the family of such geometries. Three applications will be considered: (1) Shifting estimates for eigenvalues, including ones in gaps of the essential spectrum, as well as shifting estimates for edges of components of essential spectrum, under the influence of a semidefinite potential (2) Anderson localization for general classes of random potentials with small support, and (3) null-controllability of the heat equation with interior control.

• Uncertainty relations and a sharp criterion for controllability of the heat equation., 7. January.
• Approximating the integrated density of states of random Schrödinger operators - results from empirical process theory., TU Dortmund, online via Zoom, June 1, digital.
• Logvinenko-Sereda type inequality on the sphere., TU Dortmund, June 1, digital.
• On a minimax principle in spectral gaps., University of Zagreb, Croatia, April 26, digital.

This talk deals with a minimax principle for eigenvalues in gaps of the essential spectrum of perturbed self-adjoint operators. This builds upon an abstract minimax principle devised by Griesemer, Lewis, and Siedentop and recents developments on block diagonalization of operators and forms in the off-diagonal perturbation setting. The Stokes operator is revisited as an example.

• On a minimax principle in spectral gaps., Dortmund, March 22–25, slides only. [URL]

This talk deals with a minimax principle for eigenvalues in gaps of the essential spectrum of perturbed self-adjoint operators. This builds upon an abstract minimax principle devised by Griesemer, Lewis, and Siedentop and recents developments on block diagonalization of operators and forms in the off-diagonal perturbation setting. The Stokes operator is revisited as an example.

2020
• Uncertainty relations, control theory and perturbation of spectral bands. Analysis and/of PDE Seminar, Department of Mathematical Sciences, University of Durham, England, 5. November , 15:00 CET.

Slides of the talk are provided <a href='Veselic-Durham-2020-11-04.pdf'>here</a>.

• Quantitative unique continuation estimates and resulting uncertainty relations for Schroedinger and divergence type operators., Department of Mathematical Sciences, University of Bath, 12. October , 17:15–19:45 CEST.

The talk is devoted to quantitative unique continuation estimates and resulting uncertainty relations of solutions of elliptic differential equations and eigenfunctions of associated differential operators, as well as linear combinations thereof. Such results have recently been successfully applied in several fields of mathematical physics and applied analysis: control theory, spectral engineering of eigenvalues in band gaps, and Anderson localization for random Schroedinger operators. In this talk we will focus on properties of functions in spectral subspaces of Schroedinger operators. At the end we will give some results which apply to more general elliptic second order differential equations.

• Scale-free uncertainty relations and applications in spectral and control theory., Brijuni, Croatia, 15. September.

We present an uncertainty relation for spectral projectors of Schroedinger operators on bounded and unbounded domains. These have sevaral applications, among others in the spectral theory of random Schroedinger operators. Here we will present two applications which are likely to be of interest to the audience of the conference: Shifting of bands of the essential spectrum and of eigenvalues of Schroedinger operators and controllability of the heat equation.

• Scale-free uncertainty relations for spectral projectors and applications., Institut für Mathematik der Humboldt-Universität zu Berlin, 5. March.
• On a minimax principle in spectral gaps., Minisymposium Spectral theory of operators and matrices and partial differential equations, Chemnitz, September 14–17, digital.

This talk deals with a minimax principle for eigenvalues in gaps of the essential spectrum of perturbed self-adjoint operators. This builds upon an abstract minimax principle devised by Griesemer, Lewis, and Siedentop and recents developments on block diagonalization of operators and forms in the off-diagonal perturbation setting. The Stokes operator is revisited as an example.

• Eigenvalue Lifting for Divergence-Type Operators., Minisymposium Spectral theory of operators and matrices and partial differential equations, Chemnitz, September 14–17, digital.

This talk deals with eigenvalue lifting for divergence-type operators which describes the phenomenon that certain eigenvalues are strictly increasing when the second order term is perturbed by some non-negative function with small support. Applications include, e.g., the theory of random divergence-type operators. Since here, the random perturbation affects the coefficients of the second order term, one needs exact knowledge of the dependence on some parameters which are less relevant when working with additive random potentials. The results discussed build upon recent joint work with Ivan Veselić.

• Anderson localization beyond regular Floquet eigenvalues., Sochi, Russia, February 3–7.

We prove that Anderson localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $L^2(\mathbb{R}^d)$ is universal. By this we mean that Anderson localization holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. This talk is based on joint work with M. Täufer.

2019
• Upper and lower Lipschitz bounds for shifting the edges of the essential spectrum of Schroedinger operators., Université Côte d'Azur / Laboratoire Mathématiques & Interactions J.A. Dieudonné, 28. November.

The spectrum of periodic Schroedinger operators is well known to consist of bands of essential spectrum separated by gaps, which belong to the resolvent set. The periodicity assumption allows to exhibit much more delicate properties of the spectrum, e.g. it is purely absolutely continuous. In this talk we consider the situation that the Schroedinger operator exhibits several bands of essential spectrum, but that no periodicity is assumed. This allows then for eigenvalues in the intervals between essential spectrum components. We study how the edges of the essential spectrum (and the eigenvalues in essential gaps) are shifted when a semi-definite potential is added. Crucial ingredients in the proof are a scale-free uncertainty relation and variational principles for eigenvalues in gaps of the essential spectrum.

• Upper and lower Lipschitz bounds for perturbation of the edges of the essential spectrum., Sveučilište u Dubrovniku, 29. August.

Periodic Schroedinger operators have spectrum consisting of closed intervals as connected components. These are called spectral bands. They correspond to energies where transport is possible in the medium modelled by the Schroedinger operator. For this reason it is of interest to study perturbation of spectral bands. On the one hand, one wants to establish that for small perturbations the band will not move too much. On the other hand, for perturbations with fixed sign it possible to ensure that band edges will indeed move by a quantifiable amount. This makes spectral engineering possible. We report on such results based on unique continuation principles and variational principles for eigenvalues in gaps of the essential spectrum.

• Spectral inequalities and null control for the heat conduction problem on domains with multiscale structure., Universität Bonn, 5. July.

I discuss uncertainty relations (aka spectral inequalities) for the Laplace and Schroedinger operators on bounded and unbounded domains. The subset of observation is assumed to be a thick or an equi-distrubuted set. A new result on the control cost allows to apply the first mentioned results and study the behaviour of the control cost in several asymptotic regimes, both regarding time and geometry.

• Uncertainty relations and null control for the heat conduction problem on domains with multiscale structure., Fern-Universität Hagen, 24. June–27. June.

I discuss uncertainty relations (aka spectral inequalities) for the Laplace and Schroedinger operators on bounded and unbounded domains. The subset of observation is assumed to be a thick or an equi-distrubuted set. A new result on the control cost allows to apply the first mentioned results and study the behaviour of the control cost in several asymptotic regimes, both regarding time and geometry. Methodical analogies to the study of random Schroedinger operators are highlighted.

• Upper and lower Lipschitz bounds for perturbation of the edges of the essential spectrum. Oberseminar Mathematische Physik, Fern-Universität Hagen, 2. April.
• Null-controllability of the heat equation on bounded and unbounded domains. Oberseminar Numerische Analysis und Optimierung, TU Dortmund, 14. March.
• Approximating the integrated density of states of random Schrödinger operators., Universität Dresden, December 4.
• Approximating the integrated density of states of random Schrödinger operators - results from empirical process theory., Universität Potsdam, November 13.
• Approximation der integrierten Zustandsdichte zufälliger Schrödingeroperatoren - Resultate aus der Theorie der empirischen Prozesse., HTW Dresden, November 11.
• Approximating the integrated density of states of random Schrödinger operators - results from empirical process theory., TU Chemnitz, November 6.
• Error estimates for the uniform approximation of the integrated density of states., Sveti Martin na Muri, Croatia, September 8–13, poster.
• Unique continuation for the gradient and applications., TU Dresden, December 5.
• Wegner estimates for random divergence-type operators., TU Chemnitz, December 4.
• The reflection principle in the control problem of the heat equation., Sveti Martin na Muri, Croatia, September 8–13.

We consider the control problem for the generalized heat equation for a Schrödinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts. Moreover, we show that the corresponding control cost does not exceed the one on the whole domain. This talk is based on joint work with M. Egidi.

• Unique continuation for the gradient and applications., Sveti Martin na Muri, Croatia, September 8–13, poster. [URL]

We present a unique continuation estimate for the gradient of eigenfunctions of $H = −\mathrm{div}A\nabla$, where $A(x)$ is a symmetric, uniformly elliptic matrix. This allows us to derive a Wegner estimate for random divergence type operators of the form $H_\omega = −\mathrm{div}(1 + V_\omega )\mathrm{Id}\nabla$. Here $V_\omega$ is some appropriately chosen, non-negative random field with small support.

• Zufällige Divergenz-Typ Operatoren., TU Dortmund, May 21.

Wir betrachten zufällige Operatoren der Form $H_\omega = −\mathrm{div}(1 + V_\omega )\mathrm{Id}\nabla$. Dabei ist $V_\omega$ ein geeignet gewähltes, nicht-negatives, zufälliges Potential mit kleinem Träger.

• The reflection principle in the control problem of the heat equation., Minisymposium Control of Partial Differential Equations, Gaeta, Italy, May 20–24.

We consider the control problem for the generalized heat equation for a Schrodinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts. Moreover, we show that the corresponding control cost does not exceed the one on the whole domain. This talk is based on joint work with M. Egidi.

• Lokalisierung an Bandkanten für nicht-ergodische zufällige Schrödingeroperatoren., TU Dortmund, May 14.
2018
• Wegner estimate for Landau-breather Hamiltonians., Pontificia Universidad Catolica de Chile, 13. December.

I discuss Landau Hamiltonians with a weak coupling random electric potential of breather type. Under appropriate assumptions a Wegner estimate holds. It implies the Hölder continuity of the integrated density of states. The main challenge is the problem how to deal with non-linear dependence on the random parameters.

• Scale free unique continuation estimates with three applications., Pontificia Universidad Catolica de Chile, 06. December.

I will present scale free unique continuation estimates for functions in the range of any compact spectral interval of a Schroedinger operator on generalized parallelepipeds. The latter could be cubes, halfspaces, octants, strips, slabs or the whole space. The sampling set is equidistributed. The unique continuation estimates are very precise with respect to the energy, the potential, the coarsenes scale, the radius defining the equidistributed set and actually optimal in some of these parameters. Such quantitative unique continuation estimates are sometimes called uncertainty relations or spectral inequalities, in particular in the control theory community. These estimates have range of applications. I will present three. The first concerns lifting of edges of components of the essential spectrum, the second Wegner estimates for a variety of random potentials, and the last one control theory of the heat equation. The talk is based on joint works with Nakic, Taeufer and Tautenhahn, and loosely related with works with Egidi and Seelmann.

• Uncertainty principles and null-controllability of the heat equation on bounded and unbounded domains., Solaris, Sibenik, Croatia, 17. September–20. September.

In the talk I discuss several uncertainty relations for functions in spectral subspaces of Schrödinger operators, which can be formulated as (stationary) quantitative observability estimates. Of particular interest are unbounded domains or (a sequence of) bounded domains, with multi-scale structure and large diameter. The stationary observability estimates can be turned into control cost estimates for the heat equation, implying in particular null-controlability. The interesting question in the context of unbounded domains is: Which geometric properties needs a observability set to have in order to ensure null-controlability and efficient control cost estimates? The talk is based on two joint projects, one with I. Nakić, M. Täufer, and M. Tautenhahn, the other with M. Egidi.

• Null controllability for the heat equation. Splitsko matematicko drustvo, University of Split, 14. September.

In the talk I discuss several uncertainty relations for functions in spectral sub- spaces of Schrödinger operators, which can be formulated as (stationary) quanti- tative observability estimates. Of particular interest are unbounded domains or (a sequence of) bounded domains, with multi-scale structure and large diameter. The stationary observability estimates can be turned into control cost estimates for the heat equation, implying in particular null-controllability. In particular, I will discuss sufficient and — in the case of the pure heat equation actually — sharp geometric criteria for null-controllability. The talk is based on joint projects with M. Egidi, A. Seelmann, I. Nakić, M. Täufer, and M. Tautenhahn.

• Upper and lower Lipschitz bounds for the perturbation of edges of the essential spectrum., Universitaet Bielefeld, 01. June.

Let A be a selfadjoint operator,B a bounded symmetric operator and A+tB a perturbation. I will present upper and lower Lipschitz bounds on the function of t which locally describes the movement of edges of the essential spectrum. Analogous bounds apply also for eigenvalues within gaps of the essential spectrum. The bounds hold for an optimal range of values of the coupling constant t. This result is applied to Schroedinger operators on unbounded domains which are perturbed by a non-negative potential which is mostly equal to zero. Unique continuation estimates nevertheless ensure quantitative bounds on the lifting of spectral edges due to this semidefinite potential. This allows to perform spectral engineering in certain situations. The talk is based on <a href='https://arxiv.org/abs/1804.07816'>this preprint</a>.

• Uniform Approximation of the integrated density of states on amenable groups., Bergische Universitaet Wuppertal, 05. May.

The integrated density of states is the cumulative distribution function of the spectral measure of a random ergodic Hamiltonian. It can be approximated by cumulative distribution functions associated to finite volume Hamiltonians. We discuss uniform convergence for this approximation in the case where the Hamiltonian is defined on an Euclidean lattice, or more generally, on a discrete amenable group. We present a Banach space valued Ergodic Theorem, a infinite dimensional version of the Glivenko--Cantelli Theorem, and explicit convergence estimates for the finite volume approximations.

• Necessary and sufficient geometric condition for null-controllability of the heat equation on $\Bbb R^d$.. Workshop on "Control theory of infinite-dimensional systems", Fernuni Hagen, 10. January–12. January.

In this talk we discuss the control problem for the heat equation on $\Bbb R^d, d \geq 1$, with control set $\omega \subset \Bbb R^d$. We provide a sufficient and necessary condition (called $(\Gamma, a)$-thickness) on $\omega$ such that the heat equation is null-controllable. We give an estimate of the control cost with explicit dependency on the characteristic geometric parameters of the control set. Finally, we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. We show that the control cost estimate consistent with the $\Bbb R^d$ case. (This is joint work with Michela Egidi.)

• Approximation durch Ausschöpfungen für das Kontrollproblem der Wärmeleitungsgleichung auf unbeschränkten Gebieten., TU Dortmund, November 20.
• A minimax principle in spectral gaps., TU Hamburg-Harburg, June 28.

In [Doc. Math. 4 (1999),275--283], Griesemer, Lewis, and Siedentop devised an abstract minimax principle for eigenvalues in spectral gaps of perturbed self-adjoint operators. We show that this minimax principle can be adapted to the particular situation of bounded additive perturbations with hypotheses that are easier to check in this case. The latter is demonstrated in the framework of the Davis-Kahan $\sin2\Theta$ theorem. This talked is based on joint work with I. Nakić, M. Täufer, M. Tautenhahn, and I. Veselić.

• A minimax principle in spectral gaps., Bergische Universität Wuppertal, June 23.

In [Doc. Math. 4 (1999),275--283], Griesemer, Lewis, and Siedentop devised an abstract minimax principle for eigenvalues in spectral gaps of perturbed self-adjoint operators. We show that this minimax principle can be adapted to the particular situation of bounded additive perturbations with hypotheses that are easier to check in this case. The latter is demonstrated in the framework of the Davis-Kahan $\sin2\Theta$ theorem. This talked is based on joint work with I. Nakić, M. Täufer, M. Tautenhahn, and I. Veselić.

• A minimax principle in spectral gaps. Oberseminar Stochastik/Mathematische Physik, University of Hagen, April 11.

In [Doc. Math. 4 (1999),275--283], Griesemer, Lewis, and Siedentop devised an abstract minimax principle for eigenvalues in spectral gaps of perturbed self-adjoint operators. We show that this minimax principle can be adapted to the particular situation of bounded additive perturbations with hypotheses that are easier to check in this case. The latter is demonstrated in the framework of the Davis-Kahan $\sin2\Theta$ theorem. This talked is based on joint work with I. Nakić, M. Täufer, M. Tautenhahn, and I. Veselić.

2017
• Glivenko Cantelli and Ergodic Theorem on groups., Ruhr Uni Bochum, 23. October.
• Glivenko Cantelli and Banach-space ergodic theorems applied to the uniform approximation of the integrated density of states., Universität Trier, 25. September–29. September.

The integrated density of states is the cumulative distribution function of the spectral measure of a random ergodic Hamiltonian. It can be approximated by cumulative distribution functions associated to finite volume Hamiltonians. We study uniform convergence for this approximation in the case where the Hamiltonian is defined on an Euclidean lattice, or more generally, on a discrete amenable group. We obtain a convergence estimate which can be seen as a special case of a Banach space valued Ergodic Theorem. Our proof relies on multivariate Glivenko-Cantelli Theorems. (This is joint work with Christoph Schumacher and Fabian Schwarzenberger.)

• Glivenko-Cantelli Theory, Banach-space valued Ergodic Theorems and uniform approximation of the integrated density of states., Universität Potsdam, 31. July–04. August.

The integrated density of states is the cumulative distribution function of the spectral measure of a random ergodic Hamiltonian. It can be approximated by cumulative distribution functions associated to finite volume Hamiltonians. We study uniform convergence for this approximation in the case where the Hamiltonian is defined on an Euclidean lattice, or more generally, on a discrete amenable group. We obtain a convergence estimate which can be seen as a special case of a Banach space valued Ergodic Theorem. Our proof relies on multivariate Glivenko-Cantelli Theorems. (This is joint work with Christoph Schumacher and Fabian Schwarzenberger.)

• Unique continuation estimates and lifting of eigenvalues.., Euler Institute, Saint-Petersburg, Russia, 03. July–06. July.

Using Carleman estimates we prove scale free unique continuation estimates on bounded and unbounded domains and apply them to the spectral theory of Schroedinger operators. Inparticluar, we present eigenvalue lifting estimates and lifting estimates for spectral band edgesof periodic and similar Schroedinger operators. This is joint work with I. Nakic, M. Taeufer, and M. Tautenhahn.

• Dichotomy for the expansion of the deterministic spectrum of random Schroedinger operators., Bergische Universität Wuppertal, 2. May.
• Unique continuation estimates and the Logvinenko Sereda Theorem.. Oberseminars Stochastik/Mathematische Physik, Fernuni Hagen, 15. February.

Unique continuation estimates for solutions of partial differential equations are a topic of classical interest. More recently they have turned out to have important applications for Schroedinger operators modelling condensed matter. We will present a scale-free unique continuation estimate which is tailored for such applications. Holomorphic functions exhibit unique continuation properties as well, even more precise ones. This motivates the question, to what extent UCP for solutions of PDEs can be raised to the same level as UCP for holomorphic functions. We give some partial results in this direction.

• A critical example in the subspace perturbation problem., Trier, September 25–29.

The variation of closed subspaces associated with isolated components of the spectrum of linear self-adjoint operators under a bounded off-diagonal perturbation is considered. This is studied in terms of the difference of the corresponding orthogonal projections. Although the situation is quite well understood under certain additional assumptions on the spectrum of the unperturbed operator, the general case still poses a lot of unsolved questions. We discuss a finite dimensional example indicating that the general case indeed has a different nature than the situation with the additional spectral assumptions.

• On the subspace perturbation problem., Opatija, Croatia, September 10–15.

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

• Invariant graph subspaces and block diagonalization. Oberseminar Angewandte Analysis und Mathematische Physik, LMU München, July 26.

The problem of decomposition for unbounded self-adjoint $2\times 2$ block operator matrices by a pair of orthogonal graph subspaces is discussed. As a byproduct of our consideration, a new block diagonalization procedure is suggested that resolves related domain issues. The results are discussed in the context of a two-dimensional Dirac-like Hamiltonian. The talk is based on joint work with Konstantin A. Makarov and Stephan Schmitz.

• Invariant graph subspaces and block diagonalization., TU Dortmund, January 17.

The problem of decomposition for unbounded self-adjoint $2\times 2$ block operator matrices by a pair of orthogonal graph subspaces is discussed. As a byproduct of our consideration, a new block diagonalization procedure is suggested that resolves related domain issues. The results are discussed in the context of a two-dimensional Dirac-like Hamiltonian. The talk is based on joint work with Konstantin A. Makarov and Stephan Schmitz.

2016
• Unique continuation estimates and the Logvinenko Sereda Theorems., Universite de Strasbourg, 13. December.

Unique continuation estimates for solutions of partial differential equations are a topic of classical interest. More recently, they have turned out to have important applications for Schroedinger operators modelling condensed matter. We will present a scale-free unique continuation estimate that is tailored for such applications. Holomorphic functions exhibit unique continuation properties as well, even more precise ones. This motivates the question to what extent UCP for solutions of PDEs can be raised to the same level as UCP for holomorphic functions. We give some partial results in this direction.

• Unique continuation principle and its absence on continuum and discrete geometries., Technische Universität Wien, 12. December.

A powerful tool in the analysis of solutions of partial differential equations are unique continuation principles. Quantitative versions play an important role in inverse problems, uniqueness theorems for linear and non-linear differential equations, and in the theory of random Schroedinger operators. On the contrary quantum graphs violate the continuation principle, giving rise to new phenomena. Certain graph Laplacians exhibit similar features.

• Uncertainty relations and applications to the Schrödinger and heat conduction equations., Johannes Gutenberg Universität Mainz, 13. September.

In four lectures we discuss unique continuation principles for various classes of functions, their relation to uncertainty principles, and their application in the analysis of certain elliptic and parabolic partial differential equations. We are in particular interested in domains and coefficient functions which have a multiscale structure as it istypical for periodic and random Schrödinger operators. The first two lectures are held by Ivan Veselic, the third by Martin Tautenhahn, and the last by Michela Egidi.

• A dummies view on compressive sensing., University of Split School of Medicine, Split, Croatia, 27. August–03. September.
• Unique continuation principle and its absence on continuum space, on lattices and on quantum graphs., WIAS Berlin, 14. July–15. July.

A powerful tool in the analysis of solutions of partial differential equations are unique continuation principles. Quantitative versions play an important role in inverse problems, uniqueness theorems for linear and non-linear differential equations, and in the theory of random Schroedinger operators. On the contrary quantum graphs violate the continuation principle, giving rise to new phenomena. Certain graph Laplacians exhibit similar features.

• Multiscale equidistribution estimates for Schr ̈odinger eigenfunctions., Department of Mathematics, Faculty of Science, University of Zagreb, Croatia, 19. June–22. June.

We present two results on scale-free quantitative unique continuation of eigenfunctions of the Schr̈odinger operator and linear combinations thereof. The first result is dueto Rojas-Molina &amp; Veselić, the generalization to linear combinations of eigenfunctions to Nakić, Taeufer, Tautenhahn, &amp; Veselić. We will sketch the proof for the case of pure eigenfunctions. It relies on Carleman estimates, three annuli inequalities and geometric covering arguments

• Quantitative uncertainty principles in harmonic analysis and mathematical physics., Department of Mathematics, University of Zagreb, Croatia, 16. June.

In harmonic analysis the uncertainty principle asserts that it is impossibe that a function as well as its Fourier transform are simultaneously compactly supported. In quantum mechanics the uncertainty principle asserts that it is impossible to measure two conjugate observables with arbitraty precision simultaneously. We present recent quantitative versions of uncertainty principles as well as their relations and applications in the theory of partial differential equations and random Schroedinger operators.

• (Miss)Verständnis der Stochastik. Seminar Stochastik, Fern-Universität Hagen, 4. January.
• On the subspace perturbation problem., Trilateral German-Russian-Ukrainian summer school, Mainz, September 4–15.

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

2015
• Weihnachtsvorlesung "(Miss)Verständnis der Stochastik"., Technische Universität Chemnitz, 3. December , 15:30 bis 17:00, Zentrales Hörsaalgebäude, Raum N112..

Weihnachtsvorlesung zu Themen der Stochastik für ein breites Publikum

• Hadamard's three line theorem and Carleman estimates (Hadamardov teorem of tri pravca i Carlemanove ocjene)., Zagreb University , Mathematics department, 30. September , 17:00.

We start with the maximum modulus principle for holomorphic functions and deduce Hadamard's three line theorem and Hadamard's three circle theorem. Then we pursue the question, which of these properties are shared by solutions of elliptic partial differential equations. Without proof we state a Carleman estimate and an interpolation inequality which follows. Applications thereof are discussed, time permitting, at the end of the talk.

• Uniform ergodic theorems and the Glivenko-Cantelli theorem (Uniformni ergodički teoremi i Glivenko Cantellijev teorem)., Zagreb University , Mathematics department, 29. September , 14:30.
• Approximation and estimation of functions based on local data (Aproksimacija i ocjena funkcija na osnovu lokalnih podataka)., Rijeka University, Department of Natural Sciences and Mathematics, 28. September , 12:00.

In many areas of mathematics and its application in other sciences one is confronted with the task of estimating or reconstructing a function based on partial local data. Of course, this will not work for all functions well. Thus one needs an restriction to an adequate class of functions. This can be mathematically modeled in many ways. Spacial statistics or complex function theory are relevant areas of mathematics which come to ones mind. We present several results on reconstruction and estimation of functions which are solutions of elliptic partial differential equations on some subset of Euclidean space. We comment also on analogous statements for functions with localized Fourier transform.

• Uncertainty relations and applications (Relacije neodređenosti i primjene)., University of Split, 18. September , 12:00.
• Compressed sensing and the Calderon problem in electrical impedance tomography., University of Split, 17. September , 11:00.
• Glivenko-Cantelli theory for almost additive functions and Banach space-valued ergodic theorems on lattices. Tag der Stochastik, Friedrich-Alexander-Universität Erlangen-Nürnberg, 10. July , 14:00.

We discuss uniform convergence of distribution functions in two different settings and the relation between the two. First we consider almost additive functions on lattice patterns with well defined frequencies. It is possible to embedd this context in the framework of ergodic theorems with Banach space-valued functions. If the ergodic system is generated by iid random variables it is natural to look at the same problem as an extension of the classical Glivenko-Cantelli Theorem.

• Uncertainty relations and Wegner estimates for random breather potentials. Workshop: Periodic and Other Ergodic Problems, Isaac Newton Institue, Cambridge, 23. March , 11:30–12:30.

We present a new scale-free, quantitative unique continuation estimate for Schroedinger operators in multidimensional space. Depending on the context such estimates are sometimes called uncertainty relations, observations inequalities or spectral inequalities. To illustrate its power we prove a Wegner estimate for Schroedinger operators with random breather potentials. Here we encounter a non-linear dependence on the random coupling constants, preventing the use of standard perturbation theory. The proofs rely on an analysis of the level sets of the random potential, and can be extended to a rather general framework.

• Uncertainty principles and spectral analysis of Schroedinger operators. Seminar Programme on Periodic and Ergodic Spectral Problems, Isaac Newton Institue, Cambridge, 12. March.
• Reconstruction and estimation of rigid functions based on local data. Geometry and Topology Seminar, Durham University, 9. March.

In many areas of mathematics and its application in other sciences one is confronted with the task of estimating or recosntruction a function based on partial data. Of course, this will not work for all functions well. Thus one needs an restriction to an adequate class of functions. This can be mathematically modeled in many ways. Spacial statistics or complex function theory are relevant areas of mathematics which come to ones mind. We present several results on reconstrucion and estimation of functions which are solutions of elliptic partial differential equations on some subset of Euclidean space. We comment also on analogous statements for solutions of finite difference equations on graphs.

• Quantitative Unique continuation estimates and Wegner bounds for random Schroedinger operators. Mathematical Physics Seminar, Department of Mathematics, Bristol University, 27. February.
• Compressed Sensing and Sparse Recovery II. Forschungsseminar Analysis, TU Chemnitz, 14. January , 15:30.
• On an estimate in the subspace perturbation problem., Bern, Switzerland, October 28–30.

We study the problem of variation of spectral subspaces for linear self-adjoint operators under an additive perturbation. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators in terms of the strength of the perturbation. In our approach, we formulate a constrained optimization problem on a finite set of parameters, whose solution gives an estimate on the norm of the difference of the corresponding spectral projections. In particular, this estimate is stronger than the one recently obtained by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389–1416].

• On the subspace perturbation problem., Orsay, France, October 5–7.

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

2014
• Compressed Sensing and Sparse Recovery I. Forschungsseminar Analysis, Stochastik und Mathematische Physik, TU Chemnitz, 16. December , 17:00.
• Mini-Course: Harmonic Analysis and Random Schrödinger Operators., Facultad de Matemáticas, Pontificia Universidad Católica, Santiago de Chile, 13. November–21. November.
• Quantitative unique continuation estimate and Wegner estimate for the standard random breather potential. Einladung zum Plenarvortrag bei der International Conference Spectral Theory and Mathematical Physics, Pontificia Universidad Catolica de Chile, November.

We present a new scale-free, quantitative unique continuation estimate for Schroedinger operators in multidimensional space. Depending on the context such estimates are sometimes called uncertainty relations, observations inequalities or spectral inequalities. To illustrate its power we prove a Wegner estimate for Schroedinger operators with random breather potentials. Here we encounter a non-linear dependence on the random coupling constants, preventing the use of standard perturbation theory. The proofs rely on an analysis of the level sets of the random potential, and can be extended to a rather general framework.

• Unique countinuation principles, uncertainty relations and observability estimates for elliptic equations with multiscale structure., Ecole des Ponts Paristech, 28. October.

We consider Schroedinger operators and related elliptic partial differential equations. Domains are large cubes in Euclidean space. We are aiming for estimates which are independent of the size of the cube, since we want to pass to the thermodynamic limit. We derive scale-free quantitative unique continuation principles for eigenfunctions, and for linear combinations thereof. They can be formulated, respectively interpreted, as uncertainty relations, observability estimates, or spectral inequalities. We indicate the applicability of these estimates in various areas of analysis of PDE.

• Scale-free uncertainty principles and Wegner estimates for random breather potentials. Seminar, Universität Paris 6, 27. October , 14:30.
• Uncertainty principles applied to observation and reconstruction of functions. Einladung zum Plenarvortrag beim meeting Mathematical Physics in Jena, Universität Jena, 17. September.

In several areas of mathematics appears the task of reconstructing, or at least estimating, a function on the basis of partial data. Often the partial data contain in formation about the Fourier transform as well as about the function itself. In this case the reconstruction or observation can be facilitated by various forms of the uncertainty principle. We discuss several classical as well as recent instances of such re sults. Thereafter we focus on the case of solutions of partial differential equations, where the uncertainty relation takes the form of a unique continuation estimate. Finally, we formulare two recently obtaied results, and discuss their application to con trol theory, perturbation of eigenvalues, and random Schrödinger operators.

• Uncertainty principles applied to observation and reconstruction of functions. Chemnitz-Zagreb Workshop on Harmonic Analysis for PDE, Applications, and related topics, TU Chemnitz, 1. July , 10:00.
• Uncertainty and unique continuation principles for the observation of eigenfunctions. Oberseminar Analysis, Geometrie und Stochastik, Universität Jena, 18. June , 17:00.
• Grenzverteilungssäatze für stochastische Modelle komplexer physikalischer Systeme. Stochastisches Kolloquium, Universität Jena, June.
• Grenzverteilungssätze für Modelle ungeordneter Systeme. Workshop Stochastik, Universität zu Köln, January.
• Eigenwert-Statistiken und -Verteilungsfunktionen. Oberseminar Stochastik, Fern-Universität Hagen, January.
• Eigenwert-Statistiken und -Verteilungsfunktionen. Oberseminar Stochastik, Universität Wuppertal, 22. January , 15:00.

Der Vortrag diskutiert Gesetze der Grossen Zahlen fuer Zufallsvariablen mit Werten in einem Funktionenraum. Dies wird genutzt um die spektrale Vertelungsfunktion (Integrierte Zusantdsdichte) als uniformen Limes von normierten Eigenwertzaehlfunktionen zu definieren. Wir gehen auf asymptotische Eigenschaften der spektralen Vertelungsfunktion ein. Daraufhin betrachten wir geeigent reskalierte Eigenwerte und zeigen, dass in gewissen Regimen die entsprechnden Punktprozesse gegen einen Poissonprozess konvergieren.

• Unique continuation and equidistribution properties for eigenfunctions of elliptic operators. Seminar Stochastic and Geometric Analysis, Universität Bonn, January , 10:00.
• On the subspace perturbation problem. Functional Analysis, Operator Theory and Applications, Workshop on the Occasion of the 90th Birthday of Professor Heinz Günther Tillmann, Mainz, October 23–25.

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

2013
• Multiscale unique continuation properties of eigenfunctions. Numerical Analysis Seminar, Universität Zagreb, October.
• Equidistribution properties of eigenfunctions and solutions of PDE. QMath12, Humboldt University of Berlin, 12. September , 10:30.
• Integralabschätzungen für Eigenfunktionen auf multiplen Skalen. Forschungsseminar Harmonische Analysis, TU Chemnitz, July.
• Equidistribution properties of eigenfunctions and solutions. Symposium Operator Semi-groups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, 3. June , 14:30.
• Alloy potentials on the lattice: (absence of ) monotonicity, regularity, and spectral averages. Conference Mathematical Physics of Disordered Systems - A Conference in Honor of Leonid Pastur, FernUniversität in Hagen, 17. May , 10:05.
2012
• Equidistribution properties of PDE-eigenfunctions. Conference Spectral Theory and Its Applications, Institute of Mathematics of Bordeaux, University Bordeaux 1, 4. October , 09:00.

In a joint paper with C. Rojas-Molina we have proven that eigenfunctions of the time-independent Schrödinger-equation on large cubes (with Dirichlet or periodic b.c.) exhibit a type of quantitative equidistribution property, which is uniformly good over arbitrary lenght scales. We present this result and discuss applications, extensions and open problems.

• Equidistribution properties of eigenfunctions of PDEs. Trilateral German-French-Russian Workshop on Asymptotic analysis and spectral theory on non-compact structures, Johannes Gutenberg-Universität Mainz, September.
• Diskussion über das Zusammenspiel von Bewegung und Zufall anhand von Beispielen. Tag der offenen Tür, Studieninfos &amp; Specials, TU Chemnitz, Professur Stochastik, 9. June , 14:00.
• Gleichverteilungseigenschaften von Lösungen von PDEs. Seminar, TU Bergakademie Freiberg, Institut für Numerische Mathematik und Optimierung, 8. June , 11:00.
• Stochastische und Geometrische Aspekte in der Spektraltheorie zufälliger Operatoren. Joint workshop of the departments of Mathematics and Informatics, TU Chemnitz, June.
• Glivenko-Cantelli-Theorems, concentration inequalities, and the IDS. Conference: Mathématiques des systémes quantiques désordonnés, Institut de Mathématiques de Jussieu, Université Paris 13, 29. May , 10:50.

The Glivenko-Cantelli Theorem states that the distribution functions of empirical measures generated by real-valued iid samples converge uniformly to the distribution function of the original measure. There exist various extensions of the Theorem to the mutivariate case and to Banach-space random variables, as well as criteria when the convergence holds in a stronger topology. We discuss the relation of the above results to concentration inequalities and applications in the spectral theory of random operators.

• Anwendungen des Satzes von Glivenko-Cantelli. Oberseminar Analysis, Geometrie und Stochastik, Universität Jena, Lehrstuhl für Analysis, 11. May , 11:15.
• Eigenschaften von Lösungen von partiellen Differentialgleichungen. Mathematisches Kolloquium, Teschnische Universität Ilmenau, Institut für Mathematik, 10. May , 17:00.
• Scale-uniform quantitative unique continuation principle. Seminar Mathematische Physik, FAU Erlangen-Nürnberg, Department Mathematik, 22. March , 10:30.
• Scale-uniform quantitative unique continuation principle. Seminar Angewandte Analysis und Numerische Mathematik, TU Graz, Institut für numerische Mathematik, 19. March , 11:00.
• Uncertainty relation: applications and methods. Seminar Numerische Mathematik und Wissenschaftliches Rechnen, Universität Zagreb, Institut für Mathematik, 15. March.
• Spectral averaging in the mathematical theory of Anderson localization. Seminar des Instituts für Physik, Universität Zagreb, Institut für Physik, 14. March.
• Scale-uniform quantitative unique continuation principle. Colloquium, Universität Zagreb, Institut für Mathematik, 13. March.
• Skaleninvariantes quantitatives Eindeutiges-Fortsetzungsprinzip und Anwendungen. Analysis Seminar, Seminar Analysis, TU Dresden, 19. January.
• On an estimate in the subspace perturbation problem., Graz, Austria, August 27–31.

We study the problem of variation of spectral subspaces for linear self-adjoint operators under an additive perturbation. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators in terms of the strength of the perturbation. In our approach, we formulate a constrained optimization problem on a finite set of parameters, whose solution gives an estimate on the norm of the difference of the corresponding spectral projections. In particular, this estimate is stronger than the one recently obtained by Albeverio and Motovilov in [arXiv:1112.0149v2 (2011)].

• On an estimate in the subspace perturbation problem., Mainz, September 10–12.

We study the problem of variation of spectral subspaces for linear self-adjoint operators under an additive perturbation. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators in terms of the strength of the per-turbation. In our approach, we formulate a constrained optimization problem on a finite set of parameters, whose solution gives an estimate on the norm of the difference of the corresponding spectral projections. In particular, this estimate is stronger than the one recently obtained by Albeverio andMotovilov in [arXiv:1112.0149v2 (2011)].

2011
• Scale-free quantitative unique continuation principle. Analysis Seminar, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 15. December.
• Low lying spectrum of randomly perturbed periodic waveguides. Seminar, Doppler Institute in Department of Theoretical Physics of the Nuclear Physics Institute, Rez, 18. November.
• Unique continuation principle and Wegner estimates. Workshop "Correlations and Interactions for Random Quantum Systems", Mathematisches Forschungsinstitut Oberwolfach, 24. October.
• Unique continuation properties of solutions of elliptic second order differential equations (1) and (2). Seminar "Partielle Differentialgleichungen und Inverse Probleme", TU Chemnitz, 21. October and 11. November.

Bei vielen Typen von (partiellen) Differentialgleichungen ist bekannt, dass einen nichttriviale Lösung nicht auf einer offenen Menge verschwinden kann. Diese Eigenschaft wird im Englischen "unique continuation property" genannt. Oftmals wird sie aus einer Carleman-Ungleichung hergeleitet. Ein wichtiger Anwendungsberich ist die Eindeutigkeit der Lösung von nichtlinearen Differentialgleichungen. In manchen Situationen ist es von Interesse, eine quantitative Version der "unique continuation property" herzuleiten. Wir diskutieren dies.

• Dynamics and spectra of the Schrödinger equation. Colloquium, Mathematical Society in Split, 21. September.
• Eigenvalue flows for operator-pencils. Mathematics seminar, Sveučilište u Splitu, Fakultet elektrotehnike, strojarstva i brodogradnje, 20. September.
• Eigenvalue flows for linear pencils of operators. Workshop on quantitative spectral theory and mathematical physics, Hotel Colentum, 16. September.
• Localisation, Lifschitz tails and Wegner estimates for random Schrödinger operators. Seminar Equipe Analyse, Institut de Mathématiques de Bordeaux, 8. September.
• Zufällige Schrödingeroperatoren mit nicht-monotoner Parameter-Abhängigkeit., Johannes Gutenberg-Universität Mainz, 28. January.
• A new estimate in the subspace perturbation problem., dedicated to the memory of M.Sh.Birman, Saint-Petersburg, Russia, July 1–6.

We study the problem of variation of spectral subspaces for bounded linear self-adjoint operators. We obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. This improves a result obtained earlier by Kostrykin, Makarov and Motovilov in [Trans. Amer. Math. Soc. (2007)]. This talk is based on a joint work with K. A. Makarov.

• A new estimate in the subspace perturbation problem., Orsay, France, May 25–27.

We study the problem of variation of spectral subspaces for bounded linear self-adjoint operators. We obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. This improves a result obtained earlier by Kostrykin, Makarov and Motovilov in [Trans. Amer. Math. Soc. (2007)]. This talk is based on a joint work with K. A. Makarov.

2010
• Geometrische und Spektrale Eigenschaften von Perkolationsclustern auf Cayleygraphen., Karlsruher Institut für Technologie, 16. November.
• Spectral properties of random Schroedinger operators with general interactions. Seminar on Partial Differential Equations and Spectral Theory, Pontificia Universidad Catolica de Chile, 30. September.
• Spectral properties and averaging for discrete alloy type potentials. Plenary talk at the International Conference "Spectral Days", Pontificia Universidad Catolica de Chile, 21. September.
• Fractional moment method for non-monotone models. QMath11 Mathematical Results in Quantum Physics, Hradec Kralove, University of Hradec Kralove, 6. September.
• Discrete alloy type models: averaging and spectral properties. Invited talk at the workshop "Analysis on Graphs and its Applications Follow-up meeting", Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 29. July.
• Spectral distribution function of percolation operators. AG Mathematische Physik, 17. June.
• A new estimate in the subspace perturbation problem., Chalmers, Gothenburg, Sweden, November 9.

We study the problem of variation of spectral subspaces for bounded linear self-adjoint operators in a (separable) Hilbert space. We obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively. In particular, recent results by Kostrykin, Makarov and Motovilov from [Trans. Amer. Math. Soc., V. 359, No. 1, 77--89] and [Proc. Amer. Math. Soc., 131, 3469--3476] are strengthened. This talk is based on a joint work with K. A. Makarov.

• A new estimate in the subspace perturbation problem., Special Session Quantitative Spectral Theory of Block Matrix Operators, Berlin, July 12–16.

We study the problem of variation of spectral subspaces for bounded linear self-adjoint operators. We obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. The result is discussed for off-diagonal perturbations. In this case this improves a result obtained earlier by Kostrykin, Makarov and Motovilov in [Trans.Amer. Math. Soc. (2007)]. This talk is based on a joint work with K. A. Makarov.

2009
• Discrete alloy type models with single site potentials of changing sign. International Congress on Mathematical Physics, Invited talk in the section on Nonrelativistic Quantum Mechanics, 4. August.
• Geometrische und Spektrale Eigenschaften von Perkolationsclustern auf Cayleygraphen. job application talk, Universität Mainz, 16. July.
• Exponentieller Abfall der Greenschen Funktion für Legierungsmodelle auf $\Bbb Z$., Universität Oldenburg, 10. July.
• Percolation clusters on Calyey graphs and their spectra. "Alp-Workshop", St.~Kathrein am Offenegg bei Graz, 5. July.
• Spektrale Eigenschaften von diskreten Legierungsmodellen. Oberseminar über Mathematische Physik, Ruhr-Universität Bochum, 1. July.
• Exponential decay of Green's function for Anderson models on $\Bbb Z$ with single-site potentials of finite support. Oberseminar Analysis, Universität Mainz, 30. June.
• Geometric and spectral properties of percolation on Cayley graphs. job application talk, Universität Paderborn, 3. April.
• Bounds on the spectral shift function and applications. Pure Mathematics Colloquium, Durham University, 16. March.
• Percolation clusters on Calyey graphs and their spectra. London Analysis and Probability Seminar, Imperial College, 12. March.
• Bounds on the spectral shift function and applications. Mathematics Seminar, University College London, 11. March.
• Geometric and spectral properties of percolation on Cayley graphs. Analysis Seminar, University of Bristol, 9. March.
2008
• Geometrische und spektrale Eigenschaften von Perkolationgraphen. job application talk, Institut für Mathematik, TU Berlin, 17. December.
• Perkolationscluster auf Cayleygraphen und deren Spektren. job application talk, Faculty of Mathematics, TU Cemnitz, 12. December.
• Percolation and spectra on Cayley-graphs. Oberwolfach-Workshop "Interplay of Analysis and Probability in Physics", 2. December.
• Geometrische und spektrale Eigenschaften von Perkolationgraphen. Forschungsseminar Alogrithmische und Diskrete Mathematik, TU Chemnitz, 26. November.
• Bounds on singular values of semigroup differences and the spectral shift function. Mathematisches Kolloquium, TU Clausthal, 19. November.
• Geometrische und spektrale Aspekte von Perkolation auf allgemeinen Graphen. job application talk, Fakultät für Mathematik und Informatik, Universität Bremen, 18. November.
• Perkolation auf homogenen Graphen. job application talk, Fakultät für Mathematik und Informatik, Fern-Universität Hagen, 24. October.
• Geometrische und spektrale Eigenschaften von Perkolationteilgraphen. Oberseminar Geometrie, TU Dortmund, 23. October.
• Perkolation auf homogenen Graphen. job application talk, School of Mathematics and Computer Science, Friedrich-Schiller-Universität of Jena, 7. October.
• Classical and Quantum percolation., Institut für Mathematik C, Technische Universität Graz, 28. July.
• Singular values of semigroup differences and applications to the spectral shift function., TU Berlin, 24. July.
• Geometrische und spektrale Eigenschaften von Perkolation auf Cayleygraphen. job application talk, Universität Tübingen, 15. July.
• Percolation on Cayley and quasi-transitive graphs. Workshop des SFB 701 "Aspects of Aperiodic Order", Universität Bielefeld, 4. July.
• Geometrische und spektrale Eigenschaften von Perkolation auf Cayleygraphen. job application talk, Ruhr-Universität Bochum, 1. July.
• Estimates on singular values and the spectral shift function, with applications. job application talk, Universität Bielefeld, 26. June.
• Spectral and geometric properties of percolation on general graphs. ESF-Conference "Operator Theory, Analysis and Mathematical Physics", Bedlewo, Polen, 16. June.
• Spectral and geometric properties of percolation on general graphs. Mathematisches Kolloquium, Institut für Mathematik, TU Clausthal, 28. May.
• Spectral and geometric properties of percolation on general graphs. Seminarreihe "Quantum Field Theory and Mathematical Physics", Institut for Theoretical Physics, Universität Hamburg, 27. May.
• Integrated density of states for Schrödinger operators on metric graphs. Workshop "Mathematical Physics and Spectral Theory", Humboldt Universiy of Berlin, 25. April.
• Wegner estimates for non-monotoneously correlated alloy type models. Oberwolfach-Workshop, Disordered Systems: Random Schrödinger Operators and Random Matrices, 25. March.
• On spectral properties of correlated and non-monotone Anderson models. German Open Conference on Probability and Statistics, Aachener Stochastiktage, 6. March.
• Spectral and geometric properties of percolation on general graphs. Seminar des SFB 701, Universität Bielefeld, 23. January.
2007
• Low energy asymptotics of percolation Laplacians on Cayley graphs. Workshop "Particle systems, nonlinear diffusions, and equilibration", Hausdorff-Institut, Bonn, 15. November.
• Perkolationsprozesse und Laplaceoperatoren auf Cayleygraphen. Kolloquium des Fachbereichs Mathematik und Informatik, Philipps-Universität Marburg, 15. August.
• Low energy asymptotics of percolation Hamiltonians on graphs. Conference "Equadiff 07", TU Wien, 6. August.
• Spectral asymptotics of percolation Laplacians on amenable Cayley graphs. Workshop on "Quantum Graphs, Their Spectra and Applications", Isaac Newton Institute, 3. April.
• Lifshitz tails for random Hamiltonians monotone in the randomness. Oberwolfach Mini-Workshop "Multiscale and variational methods in materials science and the quantum theory of solids", 12. February.
2006
• Untere Schranken an Eigenwertabstände. Chemnitzer Mathematisches Colloquium, 7. December.
• Lifshitz tails for the IDS of Schrödinger operators with random breather-type potential. International Conference "Operator Theory in Quantum Physics", Prag, 28. September.
• Lifshitz tails for Schrödinger operators with random breather-type potential. Conference "Operator Theory, Analysis and Mathematical Physics", Lund, 16. June.
• Existenz- und Stetigkeitseigenschaften der Integrierten Zustandsdichte. Chemnitzer Mathematisches Colloquium, 18. May.
• Anderson-percolation Hamiltonians and compactly supported eigenstates. Interdisciplinary Workshop "Evolution on Networks", Blaubeuren, 1. May.
• Untere Schranken an die unterste spektrale L¨cke von Hamiltonoperatoren mit singulärem Potential. Seminar des Graduiertenkollegs "Hierarchie und Symmetrie in mathematischen Modellen", RWTH Aachen, 24. April.
• Anderson-percolation Hamiltonians and compactly supported eigenstates. Strukturtheorie-Seminar, TU Graz, 20. April.
• Spectral analysis of percolation Hamiltonians. Oberwolfach Mini-Workshop on "$L^2$-Spectral Invariants and the Integrated Density of States", 24. February.
• Tiefliegendes Spektrum von zufälligen Schrödingeroperatoren. Forschungsseminar Algorithmische und Diskrete Mathematik, TU Chemnitz, 25. January.
• Spectral properties of Anderson-percolation Hamiltonians on graphs. Workshop "Aspects of Spectral Theory", Erwin Schrödinger Institut, Wien, 19. January.
• Compactly supported eigenfunctions of Hamiltonians on infinite graphs and jumps of the IDS. Seminar des SFB 701, Universität Bielefeld, 4. January.
2005
• Untere Schranken an die unterste spektrale Lücke von Hamiltonoperatoren mit singulärem Potential., University Osijek, Croatia, 19. December.
• Lower bounds on the lowest spectral gap of singular potential Hamiltonians. Symposium "Dirichlet Forms, Stochastic Analysis and Interacting Systems", Universität Bonn, 24. November.
• Lower bounds on the lowest spectral gap of singular potential Hamiltonians., Université de Cergy-Pontoise, France, 28. September.
• Spectral Analysis of Percolation Hamiltonians. Workshop "Computation and Analytic Problems in Spectral Theory", Gregynog, Wales, 28. July.
• Bounds on the spectral shift function and the density of states., Max-Plank-Institut, Leipzig, Germany, 24. May.
• Localization for alloy-type models: rigorosus results and methods., Department of Physics, Chemnitz, 13. April.
• Spectral analysis of percolation Hamiltonians., Université Paris, France, 8. March.
• Quantenmechanik ungeordneter Festkörper: das Phänomen der Lokalisierung. Kolloquium der Kroatischen Mathematischen Gesellschaft, Zagreb, 23. February.
• Spectral properties of periodic and random Schrödinger operators., University of Bath, UK, 14. February.
• Spectral properties of the quantum percolation model on graphs., University of Durham, UK, 9. February.
2004
• Bounds on the spectral shift function and the density of states., TU Dresden, November.
• Spektren von Quantenperkolationsmodellen., Universität Augsburg, 22. November.
• Spectral averaging induced by random, geometric perturbations., Doppler Institute, Prag, November.
• Spectral Analysis of Percolation Hamiltonians. QMath-9: Mathematical Results in Quantum Mechanics, Giens, September.
• Bounds on the spectral shift function and the density of states. Mathematics and Physics of Disordered Systems, Oberwolfach, May.
• Spektrale Verschiebung induziert durch ein kompakt getragenes Potential., Symposium über Analysis, Universitäat Basel, April.
• Integrated density of states for random metrics on manifolds., City University of New York, February.
2003
• Spectral properties of the quantum percolation model on graphs., Ruhr-Universität Bochum, December.
• Spectral shift induced by a compactly supported potential., Universität Mainz, December.
• Spectral shift induced by a compactly supported potential., Universität Stuttgart, November.
• Spectral shift induced by a compactly supported potential., Universität Konstanz, November.
• Spectral shift induced by a compactly supported potential., University of Alabama at Birmingham, October.
• Spectral shift induced by a compactly supported potential. CalTech, October.
• Spectral shift induced by a compactly supported potential., TU Chemnitz, July.
• Spectral shift induced by a compactly supported potential., Ruhr-Universität Bochum, July.
• Spectral shift induced by a compactly supported potential., University Zagreb, July.
• Spectral properties of quantum percolation models. Conference Applied Mathematics and Scientific Computing, Brijuni, Kroatien, 25. June.
• Spectral properties of the quantum percolation model on graphs. CalTech, March.
• Wegner estimates for indefinite potentials and inverses of Toeplitz matrices. Western States Mathematical Physics Meeting, CalTech, February.
2002
• On Wegner estimates with singular and indefinite randomness., UC Irvine, 12. December.
• Integrated density of states for random metrics on manifolds. CalTech, November.
• Integrated density of states for random metrics on manifolds., Ruhr-Universität Bochum, October.
• Existence of the density of states for single site potentials with samll support in one dimension. Workshop Between Order and Disorder, Greifswald, September.
• Wegner estimate with local continuity requirements on the coupling constants. Conference on Differential Equations and Mathematical Physics, Birmingham, USA, 26. March.
• Lokalisierung an Floquet-regulären Bandkanten., Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, January.
• Gruppoide von Neumann Algebren und die Integrierte Zustandsdichte., Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, January.
• Existenz der Zustandsdichte für indefinite Legierungspotentiale., Fakultät für Mathematik, TU Chemnitz, January.
2001
• Lipschitz continuity of integrated density of states for single site potentials with small support. Mathematical results in Quantum Mechanics, Taxco, Mexico, December.
• Integrated density of states for random operators on manifolds. Workshop Schrödinger Operators, IIMAS, UNA Mexico, 7. December.
• Wegner estimate for sparse and other generalized alloy type potentials. Workshop Schrödinger Operators, IIMAS, UNA Mexico, 5. December.
• The integrated Density of states and Wegner estimates. Workshop Schrödinger Operators, IIMAS, UNA Mexico, 3. December.
• Existence of the density of states for Anderson models with indefinite single site potentials. Workshop SFB 237, Tutzing, October.
• Integrated Density of states for random Schrödinger Operators on manifolds. Mini-Conference Random Schrödinger Operators, Tsukuba, Japan, August.
• Wegner Estimate for Indefinite Anderson Potentials. Workshop Applications of Renormalization Group Methods in Mathematical Sciences, Research Institute for Mathematical Sciences, Kyoto, July.
• Wegner estimates for alloy type potentials with changing sign., University of Osaka, July.
• Localization by disorder at Floquet-regular spectral boundaries., University of Osaka, July.
• Regularity porperties of the integrated density of states of random Schrödinger operators. Conference Applied Mathematics and Scientific Computing, Dubrovnik, Croatia, June.
• Die Wegner-Abschätzung und die gemeinsame Dichte der Anderson-Kopplungskonstanten. Conference Schrödinger Operators, Oberwolfach, June.
• Spektrale Eigenschaften von periodischen und zufälligen Schrödingeroperatoren., University Zagreb, Croatia, May.
• Wegner estimates for alloy type potentials with changing sign. Schrödinger operators, Oberwolfach, May.
• Spektrale Eigenschaften von zufälligen Schrödingeroperatoren., Fern-Universität Hagen, March.
2000
• Wegner Abschätzung für indefinite, überlappende Anderson Potentiale mit glatter Dichte. AG Mathematische Physik, Bochum, 5. October.
• Wegner Abschätzung für indefinite, überlappende Einzelpotentiale mit glatter Dichte. Workshop "Ungeordnete Systeme", Bochum, May.
• Integrated density of states for Schrödinger operators on manifolds. Conference on Differential Geometry and Quantum Physics, Berlin, May.
• Integrated density of states for Schrödinger operators on manifolds. Institut für Angewandte Mathematik, Universität Bonn, January.
1999
• Wegner Abschätzung für indefinite Anderson-Potentiale. Workshop SFB 237, Bad Honnef, September.
• Wegner estimates for alloy type potentials with changing sign., University of Lulea, Sweden, September.
• Wegner estimate for the Anderson model with indefinite potential., NTNU Trondheim, Norway, August.
• Wegner estimate for the Anderson model with indefinite potential. Conference on Differential Equations and Mathematical Physics, Birmingham, USA, May.
1998
• Random potentials causing localisation. Workshop SFB 237, Bad Honnef, September.

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