Papers by Peter Stollmann
Proc Amer Math Soc, 2005
This paper is concerned with elliptic operators on plane tessellations. We show that such an oper... more 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.
Eprint Arxiv 1003 3574, Mar 1, 2010
We study measures on the real line and present various versions of what it means for such a measu... more We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using Kotani-Remling theory, we show that the resulting operators have empty absolutely continuous spectrum if the measures are not periodic. When combined with Gordon type arguments this allows us to prove purely singular continuous spectrum for some continuum models of quasicrystals.
As an application we obtain compactness of semigroups for Schr\"odinger operators with potentials... more As an application we obtain compactness of semigroups for Schr\"odinger operators with potentials whose sublevel sets are thin at infinity.
Eigenvalue concentration bounds and localisation for multi-particle Hamiltonians
The effect of disorder (i.e. deviation from periodicity) in solid state models is of fundamental ... more The effect of disorder (i.e. deviation from periodicity) in solid state models is of fundamental importance. It has stimulated an enormous effort that has led to quite a number of results since the late 1950's. From mathematical point of view, there is still a lot to be done until basic questions can be considered as rigorously settled. Quasicrystals provide an interesting and very challenging type of disorder.
We carry out a careful study of basic topological and ergodic features of Delone dynamical system... more We carry out a careful study of basic topological and ergodic features of Delone dynamical systems. We then investigate the associated topological groupoids and in particular their representations on certain direct integrals with non constant fibres. Via non-commutative-integration theory these representations give rise to von Neumann algebras of random operators. Features of these algebras and operators are discussed. Restricting our attention to a certain subalgebra of tight binding operators, we then discuss a Shubin trace formula.
We prove a variant of Sch'nol's theorem in a general setting: for generators of strongly local Di... more We prove a variant of Sch'nol's theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures.
Documenta mathematica Journal der Deutschen Mathematiker-Vereinigung
The existence of positive weak solutions is related to spectral information on the corresponding ... more The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
We consider a two-particle quantum systems in R d with interaction and in presence of a random ex... more We consider a two-particle quantum systems in R d 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.
We establish the exponential localization in a multi-particle Anderson model in a Euclidean space... more We establish the exponential localization in a multi-particle Anderson model in a Euclidean space R d , d ≥ 1, in presence of a non-trivial shortrange interaction and a random external potential of an alloy type. Specifically, we prove all eigenfunctions with eigenvalues near the lower edge of the spectrum decay exponentially in L 2 -norm.
meter of the support of u. The same estimate holds for the singular values of V e# . The proof us... more meter of the support of u. The same estimate holds for the singular values of V e# . The proof uses Wely's asymptotic law for eigenvalues on balls, the Feynman-Kac formula for Schrodinger semigroups and exit time estimates for the Brownian motion. Denote by #(, H 2 , H 1 ) the spectral shift function of the pair of operators H 1 , H 2 . Theorem 2. Let f C c and set b = sup suppf . There exist constants K 1 , K 2 depending only on d, # and diam supp u such that (1) f(#) #(#, H 1 ) d# K 1 e b +K 2 {log(1 + #f## )} #f# The same estimate holds for #(, H 1 , H 2 ). Theorem 2 is derived from Theorem 1 using Young's inequality and [4]. It improves upon estimates established in [3]. Bounds like (1) are related to the question which operator pairs have a locally bounded spectral shift function. Negative results concerning this question can be found in [5, 7] and positive in [8]. An alloy type model is a random Schrodinger operator H # = H 0 + V # , where H 0 = -#
Lecture Notes in Computational Science and Engineering, 2006
Random Walks, Boundaries and Spectra, 2011
There has been quite some activity and progress concerning spectral asymptotics of random operato... more There has been quite some activity and progress concerning spectral asymptotics of random operators that are defined on percolation subgraphs of different types of graphs. In this short survey we record some of these results and explain the necessary background coming from different areas in mathematics: graph theory, group theory, probability theory and random operators.
Lecture Notes in Physics, 2006
We discuss recent results of ours showing that geometric disorder leads to some purely singularly... more We discuss recent results of ours showing that geometric disorder leads to some purely singularly continuous spectrum generically. This is based on a slight extension of Simons Wonderland theorem. Our approach to this theorem relies on the study of generic subsets of certain spaces of measures. In this article, we elaborate on this purely measure theoretic basis of our approach.
Communications in Mathematical Physics, 2003
Despite all the analogies with "usual random" models, tight binding operators for quasicrystals e... more Despite all the analogies with "usual random" models, tight binding operators for quasicrystals exhibit a feature which clearly distinguishes them from the former: the integrated density of states may be discontinuous. This phenomenon is identified as a local effect, due to occurrence of eigenfunctions with bounded support.
We establish exponential localization for a multi-particle Anderson model in a Euclidean space of... more We establish exponential localization for a multi-particle Anderson model in a Euclidean space of an arbitrary dimension, in presence of a non-trivial short-range interaction and an alloy-type random external potential. Specifically, we prove that all eigenfunctions with eigenvalues near the lower edge of the spectrum decay exponentially.
Spectral Theory and Analysis, 2011
We present an introduction to the framework of strongly local Dirichlet forms and discuss connect... more We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
Lifshitz asymptotics for percolation Hamiltonians
Bulletin of the London Mathematical Society, 2014
We construct an expansion in generalized eigenfunctions for Schrodinger operators on metric graph... more We construct an expansion in generalized eigenfunctions for Schrodinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
Reviews in Mathematical Physics, 2007
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Papers by Peter Stollmann