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Matthias Täufer

Publications and source records attributed to Matthias Täufer.

At least 19 recordsLinked to original sources
arXiv2026

Optimal Spectral Inequality for the Higher-Dimensional Landau Operator

We prove optimal spectral inequalities for Landau operators in full space and in arbitrary dimension. Spectral inequalities are lower bounds on the L 2 -mass of functions in spectral subspaces of finite energy when integrated over a sampling set S $\subset$ R d . Landau operators are Schr{ö}dinger operators associated with a constant magnetic field of the form (-$\nabla$ + A(x)) 2 where A is a -in case of non-vanishing magnetic field -unbounded vector potential. Our strategy relies on so-called magnetic Bernstein estimates and analyticity, adapting an approach used by Kovrijkine in the context of the Logvinenko-Sereda theorem. We generalize results previously only known in dimension d = 2. The main difficulty in dimension d $\ge$ 3 are the magnetic Bernstein inequalities which, in comparison to the twodimensional case, lead to additional complications and require more delicate estimates. Our results have immediate consequences for control theory, spectral theory and mathematical physics which we comment on.

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arXiv2025

Anderson localization in high-contrast media with random spherical inclusions

We study spectral properties of partial differential operators modelling composite materials with highly contrasting constituents, comprised of soft spherical inclusions with random radii dispersed in a stiff matrix. Such operators have recently attracted significant interest from the research community, including in the context of stochastic homogenization. In particular, it has been proved that the spectrum of these operators may feature a band-gap structure in the regime where heterogeneities take place on a sufficiently small scale. However, the nature of the limiting (as the small scale tends to zero) spectrum in the above setting is non-classical and not completely understood. In this paper we prove for the first time that Anderson localization occurs near band edges, thus shedding light on the limiting spectral behaviour. Our results rely on recent nontrivial advancements in quantitative unique continuation for PDEs, in combination with assumptions on the model that are standard in the Anderson localization literature, and which we plan to relax in future works.

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arXiv2025

On The Spectrum Of Infinite Quantum Graphs

We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new criterion for compactness of the resolvent and apply this to identify a transition from purely discrete to non-empty essential spectrum among a class of infinite metric graphs, a phenomenon that seems to have no known counterpart for Laplacians on Euclidean domains of infinite volume. In the case of discrete spectrum we then prove upper and lower bounds on eigenvalues, thus extending a number of bounds previously only known in the compact setting to infinite graphs. Some of our bounds, for instance in terms of the inradius, are new even on compact graphs.

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arXiv2025

Graph structure of the nodal set and bounds on the number of critical points of eigenfunctions on Riemannian manifolds

In this article, we illustrate and draw connections between the geometry of zero sets of eigenfunctions, graph theory and the vanishing order of eigenfunctions. We identify the nodal set of an eigenfunction of the Laplacian (with smooth potential) on a compact, two-dmensional Riemannian manifolds, that is on Riemannian surfaces, as an embedded metric graph and then use tools from elementary graph theory in order to estimate the number of critical points in the nodal set of the $k$-th eigenfunction and the sum of vanishing orders at critical points in terms of $k$ and the Euler-Poincaré characteristic of the surface.

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arXiv2025

Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator

We prove a spectral inequality for the Landau operator. This means that for all $f$ in the spectral subspace corresponding to energies up to $E$, the $L^2$-integral over suitable $S \subset \mathbb{R}^2$ can be lower bounded by an explicit constant times the $L^2$-norm of $f$ itself. We identify the class of all measurable sets $S \subset \mathbb{R}^2$ for which such an inequality can hold, namely so-called thick or relatively dense sets, and deduce an asymptotically optimal expression for the constant in terms of the energy, the magnetic field strength and in terms of parameters determining the thick set $S$. Our proofs rely on so-called magnetic Bernstein inequalities. As a consequence, we obtain the first proof of null-controllability for the magnetic heat equation (with sharp bound on the control cost), and can relax assumptions in existing proofs of Anderson localization in the continuum alloy-type model.

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arXiv2025

Faber-Krahn inequality for the heat content on quantum graphs via random walk expansion

We study the heat content on quantum graphs and investigate whether an analogon of the Rayleigh-Faber-Krahn inequality holds. This means that heat content at time $T$ among graphs of equal volume would be maximized by intervals (the graph analogon of balls as in the classic Rayleigh-Faber-Krahn inequality). We prove that this holds at extremal times, that is at small and at large times. For this, we employ two complementary approaches: In the large time regime, we rely on a spectral-theoretic approach, using Mercer's theorem whereas the small-time regime is dealt with by a random walk approach using the Feynman-Kac formula and Brownian motions on metric graphs. In particular, in proving the latter, we develop a new expression for the heat content as a positive linear combination of expected return times of (discrete) random walks - a formulation which seems to yield additional insights compared to previously available methods such as the celebrated Roth formula and which is crucial for our proof. The question whether a Rayleigh-Faber-Krahn inequality for the heat content on metric graphs holds at all times remains open.

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arXiv2024

The curious spectra and dynamics of non-locally finite crystals

This paper is devoted to the investigation of the spectral theory and dynamical properties of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These graphs are shown to exhibit rather intriguing behaviour: for example, we construct a periodic graph whose Laplacian has purely singular continuous spectrum. Regarding point spectrum, and different to the locally finite case, we construct a graph with a partly flat band whose eigenvectors must have infinite support. Concerning dynamical aspects, under some assumptions we prove that motion remains ballistic along at least one layer. We also construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. This provides a negative answer to an open question in this context. Furthermore, we include a discussion of the fractional Laplacian for which we prove a phase transition in its dynamical behaviour. Generally speaking, many questions still remain open, and we believe that the studied class of graphs can serve as a playground to better understand exotic spectra and dynamics.

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arXiv2024

Torsional Rigidity on Metric Graphs with Delta-Vertex Conditions

We investigate the torsion function or landscape function and its integral, the torsional rigidity, of Laplacians on metric graphs subject to $δ$-vertex conditions. A variational characterization of torsional rigidity and Hadamard-type formulas are obtained, enabling the derivation of surgical principles. We use these principles to prove upper and lower bounds on the torsional rigidity and identify graphs maximizing and minimizing torsional rigidity among classes of graphs. We also investigate the question of positivity of the torsion function and reduce it to positivity of the spectrum of a particular discrete, weighted Laplacian. Additionally, we explore potential manifestations of Kohler-Jobin-type inequalities in the context of $δ$-vertex conditions.

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arXiv2024

Towards a theory of eigenvalue asymptotics on infinite metric graphs: the case of diagonal combs

We examine diagonal combs, a recently identified class of infinite metric graphs whose properties depend on one parameter. These graphs exhibit a fascinating regime where they possess infinite volume while maintaining purely discrete spectrum for the Neumann Laplacian. In this regime, we establish polynomial upper and lower bounds on the $k$-th eigenvalue, revealing that the eigenvalues grow at a rate strictly slower than quadratic. However, once the diagonal combs transition to finite volume, their growth accelerates to a quadratic rate. Our methodology involves employing spectral geometric principles tailored for metric graphs, complemented by deriving estimates for the $k$-th eigenvalue on compact metric graphs.

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arXiv2023

Controllability of the Schrödinger equation on unbounded domains without geometric control condition

We prove controllability of the Schrödinger equation in $\mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets. This demonstrates in particular that controllability of the Schrödinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schrödinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation. Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.

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arXiv2023

Wegner estimate and localisation for alloy type operators with minimal support assumptions on the single site potential

We prove a Wegner estimate for alloy type models merely assuming that the single site potential is lower bounded by a characteristic function of a thick set, that is a particular set of positive measure. The proof is based on two ingredients: New unique continuation principles or uncertainty relations for linear combinations of eigenfunctions of the Laplacian on cubes from [EV20] and established proofs for Wegner estimates, e.g.~from [CHK07]. We obtain a Wegner estimate with optimal volume dependence at all energies, and localisation near the minimum of the spectrum, even for some non-stationary random potentials. We complement the result by showing that a lower bound on the potential by the characteristic function of a thick set is necessary for a (translation uniform) Wegner estimate to hold. Hence, we have identified a sharp condition on the size for the support of random potentials that is sufficient and necessary for the validity of Wegner estimates.

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arXiv2023

Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice

We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the $(3.6)^2$ Kagome lattice and the $(3.12)^2$ ``Super-Kagome'' lattice. We characterize all possible choices for edge weights which lead to flat bands. Furthermore, we discuss spectral consequences such as the emergence of new band gaps. Among our main findings is that flat bands are robust under physically reasonable assumptions on the perturbation and we completely describe the perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome lattice are shown to even exhibit an ``all-or-nothing'' phenomenon in the sense that there is no perturbation which can destroy only one flat band while preserving the other.

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arXiv2021

On the spectral gap of higher-dimensional Schrödinger operators on large domains

We study the asymptotic behaviour of the spectral gap of Schrödinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not.

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arXiv2021

On fully supported eigenfunctions of quantum graphs

We prove that every metric graph which is a tree has an orthonormal sequence of Laplace-eigenfunctions of full support. This implies that the number of nodal domains $ν_n$ of the $n$-th eigenfunction of the Laplacian with standard conditions satisfies $ν_n/n \to 1$ along a subsequence and has previously only been known in special cases such as mutually rationally dependent or rationally independent side lengths. It shows in particular that the Pleijel nodal domain asymptotics from two- or higher dimensional domains cannot occur on these graphs: Despite their more complicated topology, they still behave as in the one-dimensional case. We prove an analogous result on general metric graphs under the condition that they have at least one Dirichlet vertex. Furthermore, we generalize our results to Delta vertex conditions and to edgewise constant potentials. The main technical contribution is a new expression for a secular function in which modifications to the graph, to vertex conditions, and to the potential are particularly easy to understand.

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arXiv2020

The Statistics of Noisy One-Stage Group Testing in Outbreaks

In one-stage or non-adaptive group testing, instead of testing every sample unit individually, they are split, bundled in pools, and simultaneously tested. The results are then decoded to infer the states of the individual items. This combines advantages of adaptive pooled testing, i. e. saving resources and higher throughput, with those of individual testing, e. g. short detection time and lean laboratory organisation, and might be suitable for screening during outbreaks. We study the COMP and NCOMP decoding algorithms for non-adaptive pooling strategies based on maximally disjunct pooling matrices with constant row and column sums in the linear prevalence regime and in the presence of noisy measurements motivated by PCR tests. We calculate sensitivity, specificity, the probabilities of Type I and II errors, and the expected number of items with a positive result as well as the expected number of false positives and false negatives. We further provide estimates on the variance of the number of positive and false positive results. We conduct a thorough discussion of the calculations and bounds derived. Altogether, the article provides blueprints for screening strategies and tools to help decision makers to appropriately tune them in an outbreak.

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arXiv2020

Sharp estimates and homogenization of the control cost of the heat equation on large domains

We prove new bounds on the control cost for the abstract heat equation, assuming a spectral inequality or uncertainty relation for spectral projectors. In particular, we specify quantitatively how upper bounds on the control cost depend on the constants in the spectral inequality. This is then applied to the heat flow on bounded and unbounded domains modeled by a Schrödinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. The observability/control set is assumed to obey an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that our control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation that the control set becomes more and more evenly distributed throughout the domain while its density remains constant.

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arXiv2020

Unique continuation and lifting of spectral band edges of Schrödinger operators on unbounded domains (With an Appendix by Albrecht Seelmann)

We prove and apply two theorems: First, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schrödinger operator on a bounded or unbounded domain, second, a perturbation and lifting estimate for edges of the essential spectrum of a self-adjoint operator under a semi-definite perturbation. These two results are combined to obtain lower and upper Lipschitz bounds on the function parametrizing locally a chosen edge of the essential spectrum of a Schrödinger operator in dependence of a coupling constant. Analogous estimates for eigenvalues, possibly in gaps of the essential spectrum, are exhibited as well.

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