Talks and posters of Analysis chair members

Given a lower semibounded self-adjoint operator in an $L^2$-space, a spectral inequality relates for functions in certain spectral subspaces the $L^2$-norm on the whole domain to the $L^2$-norm on a suitable measurable subset. Of main interest is to understand the dependence of the corresponding ratio on certain model parameters. In this talk, typical dependencies with respect to the spectral parameter and the geometry of the measurable subset are discussed and certain recent results in this context are surveyed. The talk is based on joint works with A. Dicke and I. Veseli and with P. Alphonse.

Quantitative spectral inequalities for (anisotropic) Shubin operators on the whole space are discussed that exhibit an explicit dependence on decay rate of the sensor set. This extends recently established results by J. Martin and allows the application in the control problem for certain Baouendi-Grushin operators. This talk is based on joint work with P. Alphonse.

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

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

The integrated density of states is the (disorder averaged and volume normalized) distribution function of the spectrum of a stochastically spatially homogeneous Hamiltonian. An alternative definition is based on a sequence the empirical spectral distribution function of larger and larger matrices. These matrices approximate the stochastically homogeneous Hamiltonian. It is known that the empirical distribution functions converge uniformly to the above defined integrated density of states. The talk will discuss how this convergence (rate) can be made explicit.

For the heat equation on $\mathbb{R}^d$ it is known that the heat equation is observable from a sensor set if and only if the set is thick. For (sufficiently regular) bounded domains observability of the heat equation holds already if the sensor set has positive Lebesgue measure. We consider a third class of models lying between the two just mentioned and motivated by kinetic theory. The semigroup genera- tor is a Schroedinger operator with a quadratic or some other regularly growing potential. We identify classes of sensors sets leading to observability and null controllability. In particular, in some cases finite volume sensor sets are al- lowed, even though the configuration space is unbounded. This is joint work with Alexander Dicke and Albrecht Seelmann.

Random polytopes have a long history, going back to Sylvester's famous four-point problem of the 19th century. Since then their study has become a mainstream topic in convex and stochastic geometry, with close connection to polytopal approximation problems, among other things. In this talk we will consider random polytopes in constant curvature geometries, and show that their volume satisfies a central limit theorem. The proof uses Stein's method for normal approximation, and extends to general projective Finsler metrics.

The Poisson hyperplane tessellation in Euclidean space is a classical and well-studied model of Stochastic Geometry. In this talk we will consider its counterpart(s) in hyperbolic space. As it turns out, Euclidean hyperplanes have more than one analogue in hyperbolic space, corresponding, roughly speaking, to thinking of hyperplanes as either totally geodesic hypersurfaces, equidistant sets from other hyperplanes, or spheres of infinite radius. In hyperbolic space these notions give rise to a one-parametric family of hyperplane-like hypersurfaces, known collectively as $λ$-geodesic hypersurfaces, for $λ$ between $0$ and $1$. We will consider the isometry-invariant Poisson process of $λ$-geodesic hyperplanes, and in particular its total surface area within a growing spherical window. We will study its limit theory, and prove central and non-central limit theorems, depending on the value of $λ$ and the ambient dimension.

The Poisson hyperplane tessellation in Euclidean space is a classical and well-studied model of Stochastic Geometry. In this talk we will consider its counterpart(s) in hyperbolic space. As it turns out, Euclidean hyperplanes have more than one analogue in hyperbolic space, corresponding, roughly speaking, to thinking of hyperplanes as either totally geodesic hypersurfaces, equidistant sets from other hyperplanes, or spheres of infinite radius. In hyperbolic space these notions give rise to a one-parametric family of hyperplane-like hypersurfaces, known collectively as $λ$-geodesic hypersurfaces, for $λ$ between $0$ and $1$. We will consider the isometry-invariant Poisson process of $λ$-geodesic hyperplanes, and in particular its total surface area within a growing spherical window. We will study its limit theory, and prove central and non-central limit theorems, depending on the value of $λ$ and the ambient dimension.

We show that the restriction of a polynomial to a sphere satisfies a Logvinenko- Sereda-Kovrijkine type inequality. This implies a spectral inequality for the Laplace- Beltrami operator, which, in turn, yields observability and null-controllability with explicit estimates on the control costs for the spherical heat equation that are sharp in the large and in the small time regime. This is joint work with Alexander Dicke.

For the heat equation on Rd it is known that the heat equation is observable from a sensor set if and only if the set is thick. For (suffciently regular) bounded domains observability of the heat equation holds already if the sensor set has positive Lebesgue measure. We consider a third class of models motivated by kinetic theory. The semigroup generator is a Schroedinger operator with a quadratic or some other regularly growing potential. Quadratic elliptic differential operators are considered as well. We identify classes of sensors sets leading to observability and null controllability. In particular, in some cases finite volume sensor sets are allowed, even though the configuration space is unbounded. This is joint work with Alexander Dicke and Albrecht Seelmann.

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

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

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

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

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

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

The Poisson hyperplane tessellation in Euclidean space is a classical and well-studied model of Stochastic Geometry. In this talk we will consider its counterpart(s) in hyperbolic space. As it turns out, Euclidean hyperplanes have more than one analogue in hyperbolic space, corresponding, roughly speaking, to thinking of hyperplanes as either totally geodesic hypersurfaces, equidistant sets from other hyperplanes, or spheres of infinite radius. In hyperbolic space these notions give rise to a parametric family of hyperplane-like hypersurfaces, known collectively as lambda-geodesic hypersurfaces, for lambda between 0 and 1. We will consider the isometry-invariant Poisson process of lambda-geodesic hyperplanes, and in particular its total surface area within a growing spherical window. We will study its limit theory, and prove central and non-central limit theorems, depending on the value of lambda and the ambient dimension. Based on joint work with Z. Kabluchko and C. Thäle

The Poisson hyperplane tessellation in Euclidean space is a classical and well-studied model of Stochastic Geometry. In this talk we will consider its counterpart(s) in hyperbolic space. As it turns out, Euclidean hyperplanes have more than one analogue in hyperbolic space, corresponding, roughly speaking, to thinking of hyperplanes as either totally geodesic hypersurfaces, equidistant sets from other hyperplanes, or spheres of infinite radius. In hyperbolic space these notions give rise to a one-parametric family of hyperplane-like hypersurfaces, known collectively as $λ$-geodesic hypersurfaces, for $λ$ between $0$ and $1$. We will consider the isometry-invariant Poisson process of $λ$-geodesic hyperplanes, and in particular its total surface area within a growing spherical window. We will study its limit theory, and prove central and non-central limit theorems, depending on the value of $λ$ and the ambient dimension.

The Poisson hyperplane tessellation in Euclidean space is a classical and well-studied model of Stochastic Geometry. In this talk we will consider its counterpart(s) in hyperbolic space. As it turns out, Euclidean hyperplanes have more than one analogue in hyperbolic space, corresponding, roughly speaking, to thinking of hyperplanes as either totally geodesic hypersurfaces, equidistant sets from other hyperplanes, or spheres of infinite radius. In hyperbolic space these notions give rise to a parametric family of hyperplane-like hypersurfaces, known collectively as lambda-geodesic hypersurfaces, for lambda between 0 and 1. We will consider the isometry-invariant Poisson process of lambda-geodesic hyperplanes, and in particular its total surface area within a growing spherical window. We will study its limit theory, and prove central and non-central limit theorems, depending on the value of lambda and the ambient dimension. Based on joint work with Z. Kabluchko and C. Thäle

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

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

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

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

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.

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.

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.

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.

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.

Random polytopes have a long history, going back to Sylvester's famous four-point problem. Since then their study has become a mainstream topic in convex and stochastic geometry, with close connection to polytopal approximation problems, among other things. In this talk we will consider random polytopes in constant curvature geometries, and show that their volume satisfies a central limit theorem. The proof uses Stein's method for normal approximation, and extends to general projective Finsler metrics.

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

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.

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.

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.

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.

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.

Mathematical billiards are a classical and well-studied class of dynamical systems, "a mathematician’s playground". Convex caustics, which are curves to which billiard trajectories remain forever tangent, play an important role in the study of billiards. In this talk we will discuss convex caustics in the context of Minkowski billiards, in which are billiards in non-Euclidean normed planes. In this case a natural duality arises from, roughly speaking, interchanging the roles of the billiard table and the unit ball of the (dual) norm, which leads to duality of convex caustics. Based on joint work with S. Artstein-Avidan, D. Florentin, and Y. Ostrover.

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.

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.

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.

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.

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.

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.

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.

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.

The geometry of transformation groups is a central object of study in symplectic and contact geometry. In the former, the Hamiltonian group carries the famous Hofer norm, a canonical conjugation-invariant Finsler norm. In the latter, by contrast, the contactomorphism group admits no such Finsler norms, however recently many examples of norms have been discovered. In this talk we will survey recent results about the geometries of these two groups, with an emphasis on large-scale questions.

The geometry of transformation groups is a central object of study in symplectic and contact geometry. In the former, the Hamiltonian group carries the famous Hofer norm, a canonical conjugation-invariant Finsler norm. In the latter, by contrast, the contactomorphism group admits no such Finsler norms, however recently many examples of norms have been discovered. In this talk we will survey recent results about the geometries of these two groups, with an emphasis on large-scale questions.

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.

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.

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.

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.

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>.

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.

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.)

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.

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.

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.

The group of contactomorphisms, unlike its Hamiltonian counterpart, does not carry a canonical metric. Recently, starting from work of Sandon, many different metrics have been constructed on contactomorphism groups of various contact manifolds, using a variety of tools. We will present another example of such a metric and then discuss general restrictions on any such metric. Based on joint work with L. Polterovich and M. Fraser.

The group of Hamiltonian diffeomorphisms is a central object of study in syplectic topology. It represents all possible mechanical motions of classical phase space. The group carries a famous (and essentially unique) metric, the Hofer metric. In this talk we discuss the large-scale Hofer geometry of the Hamiltonian group, by studying its asymptotic cone. This is a standard tool in geometric group theory, which loosely speaking represents the group, as seen from infinitely far. We will see that for surfaces of genus at least 4, all asymptotic cones contain an embedded free group on two letters. All preliminaries will be discussed. Based on Joint work with D. Alvarez-Gavela, V. Kaminker, A. Kislev, K. Kliakhandler, A. Pavlichenko, L. Rigolli, O. Shabtai, B. Stevenson and J. Zhang.

We study convex caustics in Minkowski billiards. We show that for the Euclidean billiard dynamics in a planar smooth centrally symmetric and strictly convex body $K$, for every convex caustic which $K$ possesses, the "dual" billiard dynamics in which the table is the Euclidean unit disk and the geometry that governs the motion is induced by the body $K$, possesses a dual convex caustic. Such a pair of caustics is dual in a strong sense, and in particular they have the same perimeter, Lazutkin parameter (both measured with respect to the corresponding geometries), and rotation number. We show moreover that for general Minkowski billiards this phenomenon fails, and one can construct a smooth caustic in a Minkowski billiard table which possesses no dual convex caustic.

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.)

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.)

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.

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.

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.

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.

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.

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.

Mathematical billiards are a classical and well-studied class of dynamical systems, "a mathematician’s playground". Convex caustics, which are curves to which billiard trajectories remain forever tangent, play an important role in the study of billiards. In this talk we will discuss convex caustic in Minkowski billiards, which is the generalization of classical billiards no non-Euclidean normed planes. In this case a natural duality arises from, roughly speaking, interchanging the roles of the billiard table and the unit ball of the (dual) norm. This leads to duality of caustics in Minkowski billiards. Such a pair of caustics is dual in a strong sense, and in particular they have equal perimeters and other classical parameters. We will show that, when the norm is Euclidean, every caustic possesses a dual caustic, but in general this phenomenon fails. Based on joint work with S. Artstein-Avidan, D. Florentin, and Y. Ostrover.

Mathematical billiards are a classical and well-studied class of dynamical systems, "a mathematician’s playground". Convex caustics, which are curves to which billiard trajectories remain forever tangent, play an important role in the study of billiards. In this talk we will discuss convex caustic in Minkowski billiards, which is the generalization of classical billiards no non-Euclidean normed planes. In this case a natural duality arises from, roughly speaking, interchanging the roles of the billiard table and the unit ball of the (dual) norm. This leads to duality of caustics in Minkowski billiards. Such a pair of caustics is dual in a strong sense, and in particular they have equal perimeters and other classical parameters. We will show that, when the norm is Euclidean, every caustic possesses a dual caustic, but in general this phenomenon fails. Based on joint work with S. Artstein-Avidan, D. Florentin, and Y. Ostrover.

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.

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. 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 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.

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

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.

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.

The group of Hamiltonian diffeomorphisms is a central object of study in syplectic topology. It represents all possible mechanical motions of classical phase space. The group carries a famous (and essentially unique) metric, the Hofer metric. In this talk we discuss the large-scale Hofer geometry of the Hamiltonian group, by studying its asymptotic cone. This is a standard tool in geometric group theory, which loosely speaking represents the group, as seen from infinitely far. We will see that for surfaces of genus at least 4, all asymptotic cones contain an embedded free group on two letters. All preliminaries will be discussed. Based on Joint work with D. Alvarez-Gavela, V. Kaminker, A. Kislev, K. Kliakhandler, A. Pavlichenko, L. Rigolli, O. Shabtai, B. Stevenson and J. Zhang.

Weihnachtsvorlesung zu Themen der Stochastik für ein breites Publikum

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.

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.

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.

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.

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.

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].

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.

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.

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.

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.

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.

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.

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.

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.

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)].

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)].

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.

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.

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.

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.

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.