TU Dortmund
Fakultät für Mathematik

Talks and posters of Analysis chair members


Showing search results 1–241 of 241.
2021
  • Group testing, Steiner Systems, and Reed-Solomon Codes. Christoph Schumacher, Mittagsseminar Kombinatorik, Hagen, 2021–02–18. [URL]
  • Uncertainty principles and lifting of eigenvalues. Ivan Veselić, Stochastic Analysis Work Group Seminar, 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. Ivan Veselić, Workshop on control of dynamical systems, Dubrovnik, Croatia, 14. June , 09:00.

    More than 15 years ago it was established that control on any set of positive Lebesgue measure is sucient 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.

  • Three applications of scale uniform uncertainty relations for Schrodinger equations. Ivan Veselić, GAMM March 15-19, 2021 - Kassel, Germany, S23 Applied operator theory, 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.

  • Wegner estimates for Gaussian random potentials (under minimal assumptions). Ivan Veselić, Research Seminar Analysis, Stochastics and Mathematical Physics, TU Chemnitz, Faculty of Mathematics, 12. May , 15:00 CET.
  • Approximating the integrated density of states of random Schrödinger operators - results from empirical process theory. Max Kämper, , TU Dortmund, online via Zoom, June 1, digital.
  • Logvinenko-Sereda type inequality on the sphere. Alexander Dicke, Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, TU Dortmund, June 1, digital.
  • On a minimax principle in spectral gaps. Albrecht Seelmann, Seminar on Differential Equations and Numerical Analysis, 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. Albrecht Seelmann, Croatian German meeting on analysis and mathematical physics, 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. Ivan Veselić, 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. Ivan Veselić, Research seminar: Asymptotics, operators, and functionals, 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. Ivan Veselić, Tenth Conference on Applied Mathematics and Scientific Computing, 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. Ivan Veselić, Workshop: Analytical Modeling and Approximation Methods, Institut für Mathematik der Humboldt-Universität zu Berlin, 5. March.
  • On a minimax principle in spectral gaps. Albrecht Seelmann, DMV Jahrestagung 2020, 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. Alexander Dicke, DMV Jahrestagung 2020, 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. Albrecht Seelmann, Сonference on Spectral Theory and Mathematical Physics, 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. Ivan Veselić, Séminaire de l'équipe EDP Analyse Numérique, 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. Ivan Veselić, Seminar za primijenjenu matematiku i teoriju upravljanja, 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. Ivan Veselić, , 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. Ivan Veselić, Conference "On mathematical aspects of interacting systems in low dimension", 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. Ivan Veselić, Oberseminar Mathematische Physik, Fern-Universität Hagen, 2. April.
  • Null-controllability of the heat equation on bounded and unbounded domains. Ivan Veselić, Oberseminar Numerische Analysis und Optimierung, TU Dortmund, 14. March.
  • Approximating the integrated density of states of random Schrödinger operators. Max Kämper, AG "Functional analysis and dynamical systems", Universität Dresden, December 4.
  • Approximating the integrated density of states of random Schrödinger operators - results from empirical process theory. Max Kämper, Forschungsseminar Diskrete Spektraltheorie, Universität Potsdam, November 13.
  • Approximation der integrierten Zustandsdichte zufälliger Schrödingeroperatoren - Resultate aus der Theorie der empirischen Prozesse. Max Kämper, Stochastik-Club, HTW Dresden, November 11.
  • Approximating the integrated density of states of random Schrödinger operators - results from empirical process theory. Max Kämper, Research Seminar Analysis, Stochastics and Mathematical Physics, TU Chemnitz, November 6.
  • Error estimates for the uniform approximation of the integrated density of states. Max Kämper, 6th Najman Conference on Spectral Theory and Differential Equations, Sveti Martin na Muri, Croatia, September 8–13, poster.
  • Unique continuation for the gradient and applications. Alexander Dicke, Oberseminar Analysis, TU Dresden, December 5.
  • Wegner estimates for random divergence-type operators. Alexander Dicke, Research Seminar Analysis, Stochastics and Mathematical Physics, TU Chemnitz, December 4.
  • The reflection principle in the control problem of the heat equation. Albrecht Seelmann, 6th Najman Conference on Spectral Theory and Differential Equations, 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. Alexander Dicke, 6th Najman Conference on Spectral Theory and Differential Equations, Sveti Martin na Muri, Croatia, September 8–13, poster.

    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. Alexander Dicke, Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, 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. Albrecht Seelmann, International Conference on Elliptic and Parabolic Problems, 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. Albrecht Seelmann, Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, TU Dortmund, May 14.
2018
  • Wegner estimate for Landau-breather Hamiltonians. Ivan Veselić, Spectral Theory and PDE Seminar, 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. Ivan Veselić, International Conference "Spectral Theory and Mathematical Physics 2018", 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. Ivan Veselić, ApplMath18 Ninth Conference on Applied Mathematics and Scientific Computing, 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. Ivan Veselić, 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. Ivan Veselić, Kolloquium Mathematische Physik, 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. Ivan Veselić, Workshop Ergodic Theory, 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 $\Bbb R^d$.. Ivan Veselić, 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. Albrecht Seelmann, Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, TU Dortmund, November 20.
  • A minimax principle in spectral gaps. Albrecht Seelmann, Kolloquium für Angewandte Mathematik, 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. Albrecht Seelmann, 40. Nordwestdeutsches Funktionalanalysis Kolloquium, 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. Albrecht Seelmann, 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. Ivan Veselić, RTG 2131-Seminar, Ruhr Uni Bochum, 23. October.
  • Glivenko Cantelli and Banach-space ergodic theorems applied to the uniform approximation of the integrated density of states. Ivan Veselić, French-German meeting Aspect 17: Asymptotic Analysis and Spectral Theory, 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. Ivan Veselić, International Conference on Analysis and Geometry on Graphs and Manifolds, 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.. Ivan Veselić, 9th Birman Conference in Spectral Theory, 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. Ivan Veselić, Hagen-Wuppertal Analysis-Treffen, Bergische Universität Wuppertal, 2. May.
  • Unique continuation estimates and the Logvinenko Sereda Theorem.. Ivan Veselić, 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. Albrecht Seelmann, Aspect 17: Asymptotic Analysis and Spectral Theory, 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. Albrecht Seelmann, 5th Najman Conference on Spectral Theory and Differential Equations, 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. Albrecht Seelmann, 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. Albrecht Seelmann, Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, 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. Ivan Veselić, Seminaire d' Analyse, 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. Ivan Veselić, Workshop on Operator Theory and Indefinite Inner Product Spaces, 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. Ivan Veselić, Trilateral German-Russian-Ukrainian summer schoolon Spectral Theory, Differential Equations and Probability, 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. Ivan Veselić, Summer School of Medical Bionics for Hearing, 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. Ivan Veselić, NewMET 2016, 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. Ivan Veselić, International Workshop on PDEs: analysis and modelling Celebrating 80th anniversary of professor Nedžad Limić, 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. Ivan Veselić, 6th Croatian Mathematical Congress, 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. Ivan Veselić, Seminar Stochastik, Fern-Universität Hagen, 4. January.
  • On the subspace perturbation problem. Albrecht Seelmann, Spectral Theory, Differential Equations and Probability, 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". Ivan Veselić, Weihnachtsvorlesung am Tag der Lehre, 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). Ivan Veselić, Colloquium of the Croatian Mathematical Society, 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). Ivan Veselić, Probabilty seminar, 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). Ivan Veselić, Colloquium of the society of mathematicians and physicists, 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). Ivan Veselić, Seminar of the Split mathematical society, University of Split, 18. September , 12:00.
  • Compressed sensing and the Calderon problem in electrical impedance tomography. Ivan Veselić, Speech and Hearing Research Lab, School of Medicine, University of Split, 17. September , 11:00.
  • Glivenko-Cantelli theory for almost additive functions and Banach space-valued ergodic theorems on lattices. Ivan Veselić, 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. Ivan Veselić, 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. Ivan Veselić, Seminar Programme on Periodic and Ergodic Spectral Problems, Isaac Newton Institue, Cambridge, 12. March.
  • Reconstruction and estimation of rigid functions based on local data. Ivan Veselić, 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. Ivan Veselić, Mathematical Physics Seminar, Department of Mathematics, Bristol University, 27. February.
  • Compressed Sensing and Sparse Recovery II. Ivan Veselić, Forschungsseminar Analysis, TU Chemnitz, 14. January , 15:30.
  • On an estimate in the subspace perturbation problem. Albrecht Seelmann, Young Researchers Workshop on Spectral Theory, 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. Albrecht Seelmann, Aspect 15: Asymptotic Analysis and Spectral Theory, 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. Ivan Veselić, Forschungsseminar Analysis, Stochastik und Mathematische Physik, TU Chemnitz, 16. December , 17:00.
  • Mini-Course: Harmonic Analysis and Random Schrödinger Operators. Ivan Veselić, , 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. Ivan Veselić, 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. Ivan Veselić, , 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. Ivan Veselić, Seminar, Universität Paris 6, 27. October , 14:30.
  • Uncertainty principles applied to observation and reconstruction of functions. Ivan Veselić, 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. Ivan Veselić, 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. Ivan Veselić, Oberseminar Analysis, Geometrie und Stochastik, Universität Jena, 18. June , 17:00.
  • Grenzverteilungssäatze für stochastische Modelle komplexer physikalischer Systeme. Ivan Veselić, Stochastisches Kolloquium, Universität Jena, June.
  • Grenzverteilungssätze für Modelle ungeordneter Systeme. Ivan Veselić, Workshop Stochastik, Universität zu Köln, January.
  • Eigenwert-Statistiken und -Verteilungsfunktionen. Ivan Veselić, Oberseminar Stochastik, Fern-Universität Hagen, January.
  • Eigenwert-Statistiken und -Verteilungsfunktionen. Ivan Veselić, 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. Ivan Veselić, Seminar Stochastic and Geometric Analysis, Universität Bonn, January , 10:00.
  • On the subspace perturbation problem. Albrecht Seelmann, 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. Ivan Veselić, Numerical Analysis Seminar, Universität Zagreb, October.
  • Equidistribution properties of eigenfunctions and solutions of PDE. Ivan Veselić, QMath12, Humboldt University of Berlin, 12. September , 10:30.
  • Integralabschätzungen für Eigenfunktionen auf multiplen Skalen. Ivan Veselić, Forschungsseminar Harmonische Analysis, TU Chemnitz, July.
  • Equidistribution properties of eigenfunctions and solutions. Ivan Veselić, 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. Ivan Veselić, 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. Ivan Veselić, 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. Ivan Veselić, 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. Ivan Veselić, Tag der offenen Tür, Studieninfos &amp; Specials, TU Chemnitz, Professur Stochastik, 9. June , 14:00.
  • Gleichverteilungseigenschaften von Lösungen von PDEs. Ivan Veselić, Seminar, TU Bergakademie Freiberg, Institut für Numerische Mathematik und Optimierung, 8. June , 11:00.
  • Stochastische und Geometrische Aspekte in der Spektraltheorie zufälliger Operatoren. Ivan Veselić, Joint workshop of the departments of Mathematics and Informatics, TU Chemnitz, June.
  • Glivenko-Cantelli-Theorems, concentration inequalities, and the IDS. Ivan Veselić, 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. Ivan Veselić, Oberseminar Analysis, Geometrie und Stochastik, Universität Jena, Lehrstuhl für Analysis, 11. May , 11:15.
  • Eigenschaften von Lösungen von partiellen Differentialgleichungen. Ivan Veselić, Mathematisches Kolloquium, Teschnische Universität Ilmenau, Institut für Mathematik, 10. May , 17:00.
  • Scale-uniform quantitative unique continuation principle. Ivan Veselić, Seminar Mathematische Physik, FAU Erlangen-Nürnberg, Department Mathematik, 22. March , 10:30.
  • Scale-uniform quantitative unique continuation principle. Ivan Veselić, Seminar Angewandte Analysis und Numerische Mathematik, TU Graz, Institut für numerische Mathematik, 19. March , 11:00.
  • Uncertainty relation: applications and methods. Ivan Veselić, Seminar Numerische Mathematik und Wissenschaftliches Rechnen, Universität Zagreb, Institut für Mathematik, 15. March.
  • Spectral averaging in the mathematical theory of Anderson localization. Ivan Veselić, Seminar des Instituts für Physik, Universität Zagreb, Institut für Physik, 14. March.
  • Scale-uniform quantitative unique continuation principle. Ivan Veselić, Colloquium, Universität Zagreb, Institut für Mathematik, 13. March.
  • Skaleninvariantes quantitatives Eindeutiges-Fortsetzungsprinzip und Anwendungen. Ivan Veselić, Analysis Seminar, Seminar Analysis, TU Dresden, 19. January.
  • On an estimate in the subspace perturbation problem. Albrecht Seelmann, Spectral Theory and Differential Operators, 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. Albrecht Seelmann, Aspect 12: Asymptotic Analysis and Spectral Theory on Non-Compact Structures, 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. Ivan Veselić, Analysis Seminar, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 15. December.
  • Low lying spectrum of randomly perturbed periodic waveguides. Ivan Veselić, Seminar, Doppler Institute in Department of Theoretical Physics of the Nuclear Physics Institute, Rez, 18. November.
  • Unique continuation principle and Wegner estimates. Ivan Veselić, 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). Ivan Veselić, 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. Ivan Veselić, Colloquium, Mathematical Society in Split, 21. September.
  • Eigenvalue flows for operator-pencils. Ivan Veselić, Mathematics seminar, Sveučilište u Splitu, Fakultet elektrotehnike, strojarstva i brodogradnje, 20. September.
  • Eigenvalue flows for linear pencils of operators. Ivan Veselić, Workshop on quantitative spectral theory and mathematical physics, Hotel Colentum, 16. September.
  • Localisation, Lifschitz tails and Wegner estimates for random Schrödinger operators. Ivan Veselić, Seminar Equipe Analyse, Institut de Mathématiques de Bordeaux, 8. September.
  • Zufällige Schrödingeroperatoren mit nicht-monotoner Parameter-Abhängigkeit. Ivan Veselić, , Johannes Gutenberg-Universität Mainz, 28. January.
  • A new estimate in the subspace perturbation problem. Albrecht Seelmann, 3rd St.Petersburg Conference in Spectral Theory, 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. Albrecht Seelmann, Operator theory & boundary value problems, 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. Ivan Veselić, , Karlsruher Institut für Technologie, 16. November.
  • Spectral properties of random Schroedinger operators with general interactions. Ivan Veselić, Seminar on Partial Differential Equations and Spectral Theory, Pontificia Universidad Catolica de Chile, 30. September.
  • Spectral properties and averaging for discrete alloy type potentials. Ivan Veselić, Plenary talk at the International Conference "Spectral Days", Pontificia Universidad Catolica de Chile, 21. September.
  • Fractional moment method for non-monotone models. Ivan Veselić, QMath11 Mathematical Results in Quantum Physics, Hradec Kralove, University of Hradec Kralove, 6. September.
  • Discrete alloy type models: averaging and spectral properties. Ivan Veselić, 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. Ivan Veselić, AG Mathematische Physik, 17. June.
  • A new estimate in the subspace perturbation problem. Albrecht Seelmann, Analysis and Probability Seminar, 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. Albrecht Seelmann, IWOTA 2010: 21st International Workshop on Operator Theory and its Applications, 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. Ivan Veselić, International Congress on Mathematical Physics, Invited talk in the section on Nonrelativistic Quantum Mechanics, 4. August.
  • Geometrische und Spektrale Eigenschaften von Perkolationsclustern auf Cayleygraphen. Ivan Veselić, job application talk, Universität Mainz, 16. July.
  • Exponentieller Abfall der Greenschen Funktion für Legierungsmodelle auf $\Bbb Z$. Ivan Veselić, , Universität Oldenburg, 10. July.
  • Percolation clusters on Calyey graphs and their spectra. Ivan Veselić, "Alp-Workshop", St.~Kathrein am Offenegg bei Graz, 5. July.
  • Spektrale Eigenschaften von diskreten Legierungsmodellen. Ivan Veselić, 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. Ivan Veselić, Oberseminar Analysis, Universität Mainz, 30. June.
  • Geometric and spectral properties of percolation on Cayley graphs. Ivan Veselić, job application talk, Universität Paderborn, 3. April.
  • Bounds on the spectral shift function and applications. Ivan Veselić, Pure Mathematics Colloquium, Durham University, 16. March.
  • Percolation clusters on Calyey graphs and their spectra. Ivan Veselić, London Analysis and Probability Seminar, Imperial College, 12. March.
  • Bounds on the spectral shift function and applications. Ivan Veselić, Mathematics Seminar, University College London, 11. March.
  • Geometric and spectral properties of percolation on Cayley graphs. Ivan Veselić, Analysis Seminar, University of Bristol, 9. March.
2008
  • Geometrische und spektrale Eigenschaften von Perkolationgraphen. Ivan Veselić, job application talk, Institut für Mathematik, TU Berlin, 17. December.
  • Perkolationscluster auf Cayleygraphen und deren Spektren. Ivan Veselić, job application talk, Faculty of Mathematics, TU Cemnitz, 12. December.
  • Percolation and spectra on Cayley-graphs. Ivan Veselić, Oberwolfach-Workshop "Interplay of Analysis and Probability in Physics", 2. December.
  • Geometrische und spektrale Eigenschaften von Perkolationgraphen. Ivan Veselić, Forschungsseminar Alogrithmische und Diskrete Mathematik, TU Chemnitz, 26. November.
  • Bounds on singular values of semigroup differences and the spectral shift function. Ivan Veselić, Mathematisches Kolloquium, TU Clausthal, 19. November.
  • Geometrische und spektrale Aspekte von Perkolation auf allgemeinen Graphen. Ivan Veselić, job application talk, Fakultät für Mathematik und Informatik, Universität Bremen, 18. November.
  • Perkolation auf homogenen Graphen. Ivan Veselić, job application talk, Fakultät für Mathematik und Informatik, Fern-Universität Hagen, 24. October.
  • Geometrische und spektrale Eigenschaften von Perkolationteilgraphen. Ivan Veselić, Oberseminar Geometrie, TU Dortmund, 23. October.
  • Perkolation auf homogenen Graphen. Ivan Veselić, job application talk, School of Mathematics and Computer Science, Friedrich-Schiller-Universität of Jena, 7. October.
  • Classical and Quantum percolation. Ivan Veselić, , Institut für Mathematik C, Technische Universität Graz, 28. July.
  • Singular values of semigroup differences and applications to the spectral shift function. Ivan Veselić, , TU Berlin, 24. July.
  • Geometrische und spektrale Eigenschaften von Perkolation auf Cayleygraphen. Ivan Veselić, job application talk, Universität Tübingen, 15. July.
  • Percolation on Cayley and quasi-transitive graphs. Ivan Veselić, Workshop des SFB 701 "Aspects of Aperiodic Order", Universität Bielefeld, 4. July.
  • Geometrische und spektrale Eigenschaften von Perkolation auf Cayleygraphen. Ivan Veselić, job application talk, Ruhr-Universität Bochum, 1. July.
  • Estimates on singular values and the spectral shift function, with applications. Ivan Veselić, job application talk, Universität Bielefeld, 26. June.
  • Spectral and geometric properties of percolation on general graphs. Ivan Veselić, ESF-Conference "Operator Theory, Analysis and Mathematical Physics", Bedlewo, Polen, 16. June.
  • Spectral and geometric properties of percolation on general graphs. Ivan Veselić, Mathematisches Kolloquium, Institut für Mathematik, TU Clausthal, 28. May.
  • Spectral and geometric properties of percolation on general graphs. Ivan Veselić, 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. Ivan Veselić, Workshop "Mathematical Physics and Spectral Theory", Humboldt Universiy of Berlin, 25. April.
  • Wegner estimates for non-monotoneously correlated alloy type models. Ivan Veselić, Oberwolfach-Workshop, Disordered Systems: Random Schrödinger Operators and Random Matrices, 25. March.
  • On spectral properties of correlated and non-monotone Anderson models. Ivan Veselić, German Open Conference on Probability and Statistics, Aachener Stochastiktage, 6. March.
  • Spectral and geometric properties of percolation on general graphs. Ivan Veselić, Seminar des SFB 701, Universität Bielefeld, 23. January.
2007
  • Low energy asymptotics of percolation Laplacians on Cayley graphs. Ivan Veselić, Workshop "Particle systems, nonlinear diffusions, and equilibration", Hausdorff-Institut, Bonn, 15. November.
  • Perkolationsprozesse und Laplaceoperatoren auf Cayleygraphen. Ivan Veselić, Kolloquium des Fachbereichs Mathematik und Informatik, Philipps-Universität Marburg, 15. August.
  • Low energy asymptotics of percolation Hamiltonians on graphs. Ivan Veselić, Conference "Equadiff 07", TU Wien, 6. August.
  • Spectral asymptotics of percolation Laplacians on amenable Cayley graphs. Ivan Veselić, Workshop on "Quantum Graphs, Their Spectra and Applications", Isaac Newton Institute, 3. April.
  • Lifshitz tails for random Hamiltonians monotone in the randomness. Ivan Veselić, Oberwolfach Mini-Workshop "Multiscale and variational methods in materials science and the quantum theory of solids", 12. February.
2006
  • Untere Schranken an Eigenwertabstände. Ivan Veselić, Chemnitzer Mathematisches Colloquium, 7. December.
  • Lifshitz tails for the IDS of Schrödinger operators with random breather-type potential. Ivan Veselić, International Conference "Operator Theory in Quantum Physics", Prag, 28. September.
  • Lifshitz tails for Schrödinger operators with random breather-type potential. Ivan Veselić, Conference "Operator Theory, Analysis and Mathematical Physics", Lund, 16. June.
  • Existenz- und Stetigkeitseigenschaften der Integrierten Zustandsdichte. Ivan Veselić, Chemnitzer Mathematisches Colloquium, 18. May.
  • Anderson-percolation Hamiltonians and compactly supported eigenstates. Ivan Veselić, Interdisciplinary Workshop "Evolution on Networks", Blaubeuren, 1. May.
  • Untere Schranken an die unterste spektrale L¨cke von Hamiltonoperatoren mit singulärem Potential. Ivan Veselić, Seminar des Graduiertenkollegs "Hierarchie und Symmetrie in mathematischen Modellen", RWTH Aachen, 24. April.
  • Anderson-percolation Hamiltonians and compactly supported eigenstates. Ivan Veselić, Strukturtheorie-Seminar, TU Graz, 20. April.
  • Spectral analysis of percolation Hamiltonians. Ivan Veselić, Oberwolfach Mini-Workshop on "$L^2$-Spectral Invariants and the Integrated Density of States", 24. February.
  • Tiefliegendes Spektrum von zufälligen Schrödingeroperatoren. Ivan Veselić, Forschungsseminar Algorithmische und Diskrete Mathematik, TU Chemnitz, 25. January.
  • Spectral properties of Anderson-percolation Hamiltonians on graphs. Ivan Veselić, Workshop "Aspects of Spectral Theory", Erwin Schrödinger Institut, Wien, 19. January.
  • Compactly supported eigenfunctions of Hamiltonians on infinite graphs and jumps of the IDS. Ivan Veselić, Seminar des SFB 701, Universität Bielefeld, 4. January.
2005
  • Untere Schranken an die unterste spektrale Lücke von Hamiltonoperatoren mit singulärem Potential. Ivan Veselić, , University Osijek, Croatia, 19. December.
  • Lower bounds on the lowest spectral gap of singular potential Hamiltonians. Ivan Veselić, Symposium "Dirichlet Forms, Stochastic Analysis and Interacting Systems", Universität Bonn, 24. November.
  • Lower bounds on the lowest spectral gap of singular potential Hamiltonians. Ivan Veselić, , Université de Cergy-Pontoise, France, 28. September.
  • Spectral Analysis of Percolation Hamiltonians. Ivan Veselić, Workshop "Computation and Analytic Problems in Spectral Theory", Gregynog, Wales, 28. July.
  • Bounds on the spectral shift function and the density of states. Ivan Veselić, , Max-Plank-Institut, Leipzig, Germany, 24. May.
  • Localization for alloy-type models: rigorosus results and methods. Ivan Veselić, , Department of Physics, Chemnitz, 13. April.
  • Spectral analysis of percolation Hamiltonians. Ivan Veselić, , Université Paris, France, 8. March.
  • Quantenmechanik ungeordneter Festkörper: das Phänomen der Lokalisierung. Ivan Veselić, Kolloquium der Kroatischen Mathematischen Gesellschaft, Zagreb, 23. February.
  • Spectral properties of periodic and random Schrödinger operators. Ivan Veselić, , University of Bath, UK, 14. February.
  • Spectral properties of the quantum percolation model on graphs. Ivan Veselić, , University of Durham, UK, 9. February.
2004
  • Bounds on the spectral shift function and the density of states. Ivan Veselić, , TU Dresden, November.
  • Spektren von Quantenperkolationsmodellen. Ivan Veselić, , Universität Augsburg, 22. November.
  • Spectral averaging induced by random, geometric perturbations. Ivan Veselić, , Doppler Institute, Prag, November.
  • Spectral Analysis of Percolation Hamiltonians. Ivan Veselić, QMath-9: Mathematical Results in Quantum Mechanics, Giens, September.
  • Bounds on the spectral shift function and the density of states. Ivan Veselić, Mathematics and Physics of Disordered Systems, Oberwolfach, May.
  • Spektrale Verschiebung induziert durch ein kompakt getragenes Potential. Ivan Veselić, , Symposium über Analysis, Universitäat Basel, April.
  • Integrated density of states for random metrics on manifolds. Ivan Veselić, , City University of New York, February.
2003
  • Spectral properties of the quantum percolation model on graphs. Ivan Veselić, , Ruhr-Universität Bochum, December.
  • Spectral shift induced by a compactly supported potential. Ivan Veselić, , Universität Mainz, December.
  • Spectral shift induced by a compactly supported potential. Ivan Veselić, , Universität Stuttgart, November.
  • Spectral shift induced by a compactly supported potential. Ivan Veselić, , Universität Konstanz, November.
  • Spectral shift induced by a compactly supported potential. Ivan Veselić, , University of Alabama at Birmingham, October.
  • Spectral shift induced by a compactly supported potential. Ivan Veselić, CalTech, October.
  • Spectral shift induced by a compactly supported potential. Ivan Veselić, , TU Chemnitz, July.
  • Spectral shift induced by a compactly supported potential. Ivan Veselić, , Ruhr-Universität Bochum, July.
  • Spectral shift induced by a compactly supported potential. Ivan Veselić, , University Zagreb, July.
  • Spectral properties of quantum percolation models. Ivan Veselić, Conference Applied Mathematics and Scientific Computing, Brijuni, Kroatien, 25. June.
  • Spectral properties of the quantum percolation model on graphs. Ivan Veselić, CalTech, March.
  • Wegner estimates for indefinite potentials and inverses of Toeplitz matrices. Ivan Veselić, Western States Mathematical Physics Meeting, CalTech, February.
2002
  • On Wegner estimates with singular and indefinite randomness. Ivan Veselić, , UC Irvine, 12. December.
  • Integrated density of states for random metrics on manifolds. Ivan Veselić, CalTech, November.
  • Integrated density of states for random metrics on manifolds. Ivan Veselić, , Ruhr-Universität Bochum, October.
  • Existence of the density of states for single site potentials with samll support in one dimension. Ivan Veselić, Workshop Between Order and Disorder, Greifswald, September.
  • Wegner estimate with local continuity requirements on the coupling constants. Ivan Veselić, Conference on Differential Equations and Mathematical Physics, Birmingham, USA, 26. March.
  • Lokalisierung an Floquet-regulären Bandkanten. Ivan Veselić, , Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, January.
  • Gruppoide von Neumann Algebren und die Integrierte Zustandsdichte. Ivan Veselić, , Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, January.
  • Existenz der Zustandsdichte für indefinite Legierungspotentiale. Ivan Veselić, , Fakultät für Mathematik, TU Chemnitz, January.
2001
  • Lipschitz continuity of integrated density of states for single site potentials with small support. Ivan Veselić, Mathematical results in Quantum Mechanics, Taxco, Mexico, December.
  • Integrated density of states for random operators on manifolds. Ivan Veselić, Workshop Schrödinger Operators, IIMAS, UNA Mexico, 7. December.
  • Wegner estimate for sparse and other generalized alloy type potentials. Ivan Veselić, Workshop Schrödinger Operators, IIMAS, UNA Mexico, 5. December.
  • The integrated Density of states and Wegner estimates. Ivan Veselić, Workshop Schrödinger Operators, IIMAS, UNA Mexico, 3. December.
  • Existence of the density of states for Anderson models with indefinite single site potentials. Ivan Veselić, Workshop SFB 237, Tutzing, October.
  • Integrated Density of states for random Schrödinger Operators on manifolds. Ivan Veselić, Mini-Conference Random Schrödinger Operators, Tsukuba, Japan, August.
  • Wegner Estimate for Indefinite Anderson Potentials. Ivan Veselić, 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. Ivan Veselić, , University of Osaka, July.
  • Localization by disorder at Floquet-regular spectral boundaries. Ivan Veselić, , University of Osaka, July.
  • Regularity porperties of the integrated density of states of random Schrödinger operators. Ivan Veselić, Conference Applied Mathematics and Scientific Computing, Dubrovnik, Croatia, June.
  • Die Wegner-Abschätzung und die gemeinsame Dichte der Anderson-Kopplungskonstanten. Ivan Veselić, Conference Schrödinger Operators, Oberwolfach, June.
  • Spektrale Eigenschaften von periodischen und zufälligen Schrödingeroperatoren. Ivan Veselić, , University Zagreb, Croatia, May.
  • Wegner estimates for alloy type potentials with changing sign. Ivan Veselić, Schrödinger operators, Oberwolfach, May.
  • Spektrale Eigenschaften von zufälligen Schrödingeroperatoren. Ivan Veselić, , Fern-Universität Hagen, March.
2000
  • Wegner Abschätzung für indefinite, überlappende Anderson Potentiale mit glatter Dichte. Ivan Veselić, AG Mathematische Physik, Bochum, 5. October.
  • Wegner Abschätzung für indefinite, überlappende Einzelpotentiale mit glatter Dichte. Ivan Veselić, Workshop "Ungeordnete Systeme", Bochum, May.
  • Integrated density of states for Schrödinger operators on manifolds. Ivan Veselić, Conference on Differential Geometry and Quantum Physics, Berlin, May.
  • Integrated density of states for Schrödinger operators on manifolds. Ivan Veselić, Institut für Angewandte Mathematik, Universität Bonn, January.
1999
  • Wegner Abschätzung für indefinite Anderson-Potentiale. Ivan Veselić, Workshop SFB 237, Bad Honnef, September.
  • Wegner estimates for alloy type potentials with changing sign. Ivan Veselić, , University of Lulea, Sweden, September.
  • Wegner estimate for the Anderson model with indefinite potential. Ivan Veselić, , NTNU Trondheim, Norway, August.
  • Wegner estimate for the Anderson model with indefinite potential. Ivan Veselić, Conference on Differential Equations and Mathematical Physics, Birmingham, USA, May.
1998
  • Random potentials causing localisation. Ivan Veselić, Workshop SFB 237, Bad Honnef, September.

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