Alt, B. G artner, Seminar uber Algorithmische Geometrie Seminar on. Algorithm 3 Processing P-Nodes Input: Exaples are the complete graphs, the complete bipartite graphs, hypercubes or the Petersen graph. Lightness of digraphs in surfaces and directed game chromatic numberDiscrete Mathematics— Andres SD: Geometrische und Spektrale Eigenschaften von Perkolationsclustern auf Cayleygraphen.
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Sekretariat Janine Textor Raum M Lehre im aktuellen Semester. Illia Karabash. Christoph Schumacher.
Albrecht Seelmann. Karl Friedrich Siburg. Christopher Strothmann. Janine Textor Sekretariat. Curriculum Vitae. Michela Egidi. Dortmund-Hagen-Wuppertal Analysis Meeting. Workshop Mathematical Physics and Dynamical Systems TU Dortmund. Upper and lower Lipschitz bounds for shifting the edges of the essential spectrum of Schroedinger operators.
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. 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. 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. 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.
Methodical analogies to the study of random Schroedinger operators are highlighted. I discuss Landau Hamiltonians with a weak coupling random electric potential of breather type.
Under appropriate assumptions a Wegner estimate holds. The main challenge is the problem how to deal with non-linear dependence on the random parameters. 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. 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. Tautenhahn, the other with M. 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. 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. 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. This is joint work with Michela Egidi.
Glivenko Cantelli and Banach-space ergodic theorems applied to the uniform approximation of the integrated density of states. 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. 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. Dichotomy for the expansion of the deterministic spectrum of random Schroedinger operators. 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. We give some partial results in this direction. 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. 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.
Unique continuation principle and its absence on continuum space, on lattices and on quantum graphs. We will sketch the proof for the case of pure eigenfunctions.
It relies on Carleman estimates, three annuli inequalities and geometric covering arguments. 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.
Hadamard's three line theorem and Carleman estimates Hadamardov teorem of tri pravca i Carlemanove ocjene. 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.
Dr. Philipp Kindermann
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