# Lehrstuhl IX Analysis, Mathematische Physik & Dynamische Systeme

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Item Uncertainty principles with error term in Gelfand–Shilov spaces(2022-07-28) Dicke, Alexander; Seelmann, AlbrechtShow more In this note, an alternative approach to establish observability for semigroups based on their smoothing properties is presented. The results discussed here reproduce some of those recently obtained in [arXiv:2112.01788], but the current proof allows to get rid of several technical assumptions by following the standard complex analytic approach established by Kovrijkine combined with an idea from [arXiv:2201.02370].Show more Item On a minimax principle in spectral gaps(2022-03-03) Seelmann, AlbrechtShow more The minimax principle for eigenvalues in gaps of the essential spectrum in the form presented by Griesemer et al. (Doc Math 4:275–283, 1999) is adapted to cover certain abstract perturbative settings with bounded or unbounded perturbations, in particular ones that are off-diagonal with respect to the spectral gap under consideration. This in part builds upon and extends the considerations in the author’s appendix to Nakić et al. (J Spectr Theory 10:843–885, 2020). Several monotonicity and continuity properties of eigenvalues in gaps of the essential spectrum are deduced, and the Stokes operator is revisited as an example.Show more Item Spectral inequalities for Schrödinger operators and parabolic observability(2022) Dicke, Alexander; Veselic, Ivan; Jacob, BirgitShow more This work examines the observability of abstract Cauchy problems corresponding to generators of a strongly continuous operator semigroup. A special focus is put on the investigation of sufficient conditions on the sensor sets that ensure observability. This is done by means of new spectral inequalities and dissipation estimates. First of all, the situation of selfadjoint operators is examined. Suitable spectral inequalities are proven for selfadjoint Schrödinger operators, where the potential has either mild local singularities or is (at least in certain coordinate directions) a power growth potential. In the second case, the prime examples are the partial harmonic oscillators. While the spectral inequality is proven in the case of mild local singularities for equidistributed sensor sets, in the case of potentials with power growth it is shown that these sets may become sparse in directions where the potential grows unboundedly. In particular, the presented results allow to consider sensor sets with finite measure in certain cases, e.g., for the harmonic oscillator. Subsequently, new dissipation estimates for semigroups generated by quadratic differential operators are shown. Here the associated so-called comparison operator is a partial harmonic oscillator and its specific form is characterized by the singular space of the generator. In this context, the aforementioned spectral inequalities allow to establish observability of these semigroups from sensor sets that become sparse. At last, the situation is investigated where the range of the semigroup is contained in a Gelfand-Shilov space. Establishing an uncertainty principle with error term for functions from these spaces, the observability is also proven in this situation. The work is concluded by an appendix discussing a unique continuation principle for the gradient and its applications in the theory of random divergence-type operators.Show more Item Wegner estimate for random divergence-type operators monotone in the randomness(2021-06-19) Dicke, AlexanderShow more In this note, a Wegner estimate for random divergence-type operators that are monotone in the randomness is proven. The proof is based on a recently shown unique continuation estimate for the gradient and the ensuing eigenvalue liftings. The random model which is studied here contains quite general random perturbations, among others, some that have a non-linear dependence on the random parameters.Show more Item Extremal and functional dependence between continuous random variables(2021) Strothmann, Christopher; Siburg, Karl Friedrich; Trutschnig, WolfgangShow more Describing and measuring the dependence between random variables is crucial to inform decisions regarding investments, policies, or even public safety. As the well-known coefficient of correlation cannot discriminate between independence and dependence in nonlinear settings, many other dependence concepts and corresponding measures of dependence have been proposed in the literature. This thesis investigates different dependence concepts for continuous random variables, while a special focus is placed on two particular dependence concepts, namely tail dependence and complete dependence. In the first part of the thesis, we define the tail dependence ordering to consistently compare the degree of extremal dependence encoded in different sets of random variables and discuss corresponding measures of tail dependence, which include the well-known tail dependence coefficient as a special case. Furthermore, we provide conditions under which the tail dependence ordering is equivalent to a localized version of the usual stochastic dominance order. Afterwards, we investigate the tail behaviour of the (generalized) Markov product for copulas and introduce an analogous product structure for tail dependence functions. While this product is similar to the Markov product of copulas in several ways, for instance, regarding its algebraic properties or its connection to a class of linear operators, it exhibits a distinct reduction property allowing us to characterize the dynamic behaviour of n-fold iterates of the Markov product. In the second part, we introduce a novel approach to rearrange a copula C into a unique stochastically increasing copula C↑. By combining measures of concordance such as Spearman's ρ or Kendall's τ with this rearrangement, we construct new measures of complete dependence. These so-called rearranged dependence measures are shown to possess various advantageous properties, e.g. they fulfil the data processing inequality, and are consistent with the underlying concordance measure. Lastly, we consider the theoretical properties of stochastically increasing copulas in more detail. This class of copulas exhibits improved convergence and order properties and includes various well-known copula families such as Gaussian or extreme-value copulas. Most importantly, stochastically increasing copulas fulfil a reduction property very similar to that of tail dependence functions, thus allowing us to characterize idempotents and n-fold iterates as ordinal sums of the independence copula.Show more Item The Laplacian on Cartesian products with mixed boundary conditions(2021-03-03) Seelmann, AlbrechtShow more A definition of the Laplacian on Cartesian products with mixed boundary conditions using quadratic forms is proposed. Its consistency with the standard definition for homogeneous and certain mixed boundary conditions is proved and, as a consequence, tensor representations of the corresponding Sobolev spaces of first order are derived. Moreover, a criterion for the domain to belong to the Sobolev space of second order is proved.Show more