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Groupoids, von Neumann algebras and the integrated density of states (2007)
Journal Article
Lenz, D., Veselic, I., & Peyerimhoff, N. (2007). Groupoids, von Neumann algebras and the integrated density of states. Mathematical Physics, Analysis and Geometry, 10(1), 1-41. https://doi.org/10.1007/s11040-007-9019-2

We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstra... Read More about Groupoids, von Neumann algebras and the integrated density of states.

Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature (2006)
Journal Article
Klassert, S., Lenz, D., Peyerimhoff, N., & Stollmann, P. (2006). Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature. Proceedings of the American Mathematical Society, 134(5), 1549-1559

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... Read More about Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature.

Geometric heat comparison criteria for Riemannian manifolds (2006)
Journal Article
Karp, L., & Peyerimhoff, N. (2007). Geometric heat comparison criteria for Riemannian manifolds. Annals of Global Analysis and Geometry, 31, 115-145. https://doi.org/10.1007/s10455-006-9038-4

The main results of this article are small-time heat comparison results for two points in two manifolds with characteristic functions as initial temperature distributions. These results are based on the geometric concepts of "(essential) distance fro... Read More about Geometric heat comparison criteria for Riemannian manifolds.

Geodesics in non-positively curved plane tessellations (2006)
Journal Article
Baues, O., & Peyerimhoff, N. (2006). Geodesics in non-positively curved plane tessellations. Advances in Geometry, 6(2), 243-263. https://doi.org/10.1515/advgeom.2006.014

We introduce a natural combinatorial curvature function on the corners of plane tessellations and relate it to the global metric geometry of their corresponding edge and dual graphs. If the combinatorial curvature in the corners is non-positive then... Read More about Geodesics in non-positively curved plane tessellations.

Spherical means on compact locally symmetric spaces of non-positive curvature (2006)
Journal Article
Peyerimhoff, N. (2006). Spherical means on compact locally symmetric spaces of non-positive curvature. Forum Mathematicum, 18(3), 391-417. https://doi.org/10.1515/forum.2006.022

We consider spherical means of continuous functions on the unit tangent bundle of a compact, non-positively curved locally symmetric space and study their behavior as the radius tends to infinity. In dimension greater or equal to 2, we prove that sph... Read More about Spherical means on compact locally symmetric spaces of non-positive curvature.

The Cheeger constant of simply connected, solvable Lie groups (2004)
Journal Article
Peyerimhoff, N., & Samiou, E. (2004). The Cheeger constant of simply connected, solvable Lie groups. Proceedings of the American Mathematical Society, 132(5), 1525-1529

We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group G with Lie algebra g is h(G) = max tr(ad(H)), where the maximum is taken over all vectors H in g with norm one.

Integrated density of states for random metrics on manifolds (2004)
Journal Article
Lenz, D., Peyerimhoff, N., & Veselic, I. (2004). Integrated density of states for random metrics on manifolds. Proceedings of the London Mathematical Society, 88(3), 733-752. https://doi.org/10.1112/s0024611503014576

This paper carries over the fundamental properties of random Schroedinger operators to random Laplace-Beltrami operators, that is, Laplacians with random metrics. Namely, we (A) discuss a framework for ergodic, random operators on covering manifolds... Read More about Integrated density of states for random metrics on manifolds.

Random Schroedinger operators on manifolds (2003)
Journal Article
Lenz, D., Peyerimhoff, N., & Veselic, I. (2003). Random Schroedinger operators on manifolds. Markov processes and related fields, 9, 717-728

We consider a random family of Schroedinger operators on a cover of a compact Riemannian manifold. We present several results on their spectral theory, in particular almost sure constancy of th spectral components and existence and non-randomness of... Read More about Random Schroedinger operators on manifolds.