arXiv subjects
Publications and source records attributed to P. Stollmann.
A one parameter family of selfadjoint operators gives rise to a corresponding direct integral. We show how to use the Putnam Kato theorem to obtain a new method for the proof of a spectral averaging result.
We establish exponential localization for a two-particle Anderson model in a Euclidean space ${\mathbb R}^{d}$, $d\ge 1$, in presence of a non-trivial short-range interaction and a random external potential of the alloy type. Specifically, we prove that all eigenfunctions with eigenvalues near the lower edge of the spectrum decay exponentially in $L^2$-norm.
We consider a two-particle quantum systems in a d-dimensional Euclidean space with interaction and in presence of a random external potential (a continuous two-particle Anderson model). We establish Wegner-type estimates (inequalities) for such models, assessing the probability that random spectra of Hamiltonians in finite volumes intersect a given set.
This paper is concerned with elliptic operators on plane tessellations. We show that such an operator does not admit a compactly supported eigenfunction, if the combinatorial curvature of the tessellation is nonpositive. Furthermore, we show that the only geometrically finite, repetitive plane tessellations with nonpositive curvature are the regular $(3,6), (4,4)$ and $(6,3)$ tilings.
We consider continuum random Schrödinger operators of the type $H_ω = -Δ+ V_0 + V_ω$ with a deterministic background potential $V_0$. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of $-Δ+V_0$. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (``random tube'') in arbitrary dimension.
We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain subalgebra to discuss a Shubin trace formula.
We prove that a strong form of dynamical localization follows from a variable energy multi-scale analysis. This abstract result is applied to a number of models for wave propagation in disordered media.