arXiv subjects
Publications and source records attributed to Peter Stollmann.
We prove that two different abstract quantitative uncertainty principles are equivalent to the strict positivity of the associated abstract Schr\"o\-din\-ger operators. We also discuss the case of continuum Schr\"odinger operators, in which case our method provides an explicitly computable lower bound as well as a control theoretic application.
We consider linear control problems for the heat equation of the form $\dot f (t) = -Hf (t) + \mathbf{1}_D u (t)$, $f (0) \in \ell_2 (X,m)$, where $H$ is the weighted Laplacian on a discrete graph $(X,b,m)$, and where $D \subseteq X$ is relatively dense. We show cost-uniform $\alpha$-controllability by means of a weak observability estimate for the corresponding dual observation problem. We discuss optimality of our result as well as consequences on stabilizability properties.
After introducing a natural notion of continuous fields of locally convex spaces, we establish a new theory of strongly continuous families of possibly unbounded self-adjoint operators over varying Hilbert spaces. This setting allows to treat operator families defined on bundles of Hilbert spaces that are not locally trivial (such as e.g.~the tangent bundle of Wasserstein space), without referring to identification operators at all.
We define the speed measure $\nu$ for mappings $\gamma:I\to X$ from an interval to a metric space that are locally of bounded variation. We characterize continuity and absolute continuity of $\gamma$ in terms of $\nu$ and identify the Radon-Nikod\'ym derivative of $\nu$ with respect to Lebesgue measure as the metric speed of $\gamma$. In doing so we prove an extension of the Banach-Zaretsky theorem.
Given a strongly local Dirichlet space and $\lambda\geq 0$, we introduce a new notion of $\lambda$--subharmonicity for $L^1_\loc$--functions, which we call \emph{local $\lambda$--shift defectivity}, and which turns out to be equivalent to distributional $\lambda$--subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional $L^q$-solutions of $\Delta f\leq f$ for complete Riemannian manifolds.
We give necessary and sufficient conditions for a regular semi-Dirichlet form to enjoy a new Feller type property, which we call \emph{weak Feller property}. Our characterization involves potential theoretic as well as probabilistic aspects and seems to be new even in the symmetric case. As a consequence, in the symmetric case, we obtain a new variant of a decomposition principle of the essential spectrum for (the self-adjoint operators induced by) regular symmetric Dirichlet forms and a Persson type theorem, which applies e.g. to Cheeger forms on $\mathsf{RCD^*}$ spaces.
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet Laplacian on subsets that satisfy natural geometric conditions.
We derive new estimates for the first Betti number of compact Riemannian manifolds. Our approach relies on the Birman-Schwinger principle and Schatten norm estimates for semigroup differences. In contrast to previous works we do not require any a priori ultracontractivity estimates and we provide bounds which explicitly depend on suitable integral norms of the Ricci tensor.
We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence form operators with not necessarily continuous main part.
We review recent results about heat kernel estimates based on Kato conditions on the negative part of the Ricci curvature.
We study topological Poincaré type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory of Laplacians once a probability measure on the graph is chosen. More specifically, we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures.
We present a decomposition principle for general regular Dirichlet forms satisfying a spatial local compactness condition. We use the decomposition principle to derive a Persson type theorem for the corresponding Dirichlet forms. In particular our setting covers Laplace-Beltrami operators on Riemannian manifolds, and Dirichlet forms associated to $α$-stable processes in Euclidean space.
We prove a quantitative uncertainty principle at low energies for the Laplacian on fairly general weighted graphs with a uniform explicit control of the constants in terms of geometric quantities. A major step consists in establishing lower bounds for Dirichlet eigenvalues in terms of the geometry.
We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci curvature. Moreover, this provides quantitative statements about the cohomology group, contrary to results by Elworthy and Rosenberg.
We study a discrete Laplace operator $Δ$ on percolation subgraphs of an infinite graph. The ball volume is assumed to grow at most polynomially. We are interested in the behavior of the integrated density of states near the lower spectral edge. If the graph is a Cayley graph we prove that it exhibits Lifshitz tails. If we merely assume that the graph has an exhausting sequence with positive $δ$-dimensional density, we obtain an upper bound on the integrated density of states of Lifshitz type.
We study Schrödinger operators on $\R$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated by aperiodic subshifts.
In this paper we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank in \cite{BreuerFrank2009} in the discrete case as well as for sparse trees in the metric case.
Erratum to "From Uncertainty Principles to Wegner Estimates".