arXiv subjects

Peter Herbrich

Publications and source records attributed to Peter Herbrich.

7 recordsLinked to original sources

arXiv2018

Robin and Steklov isospectral manifolds

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin parameter. As in the sloshing problem, we can also impose mixed Dirichlet-Neumann conditions on parts of the boundary. Among the examples we exhibit are Steklov isospectral flat surfaces with boundary, planar domains with isospectral sloshing problems, and Steklov isospectral metrics on balls of any dimension greater than 5. In particular, the latter are the first examples of Steklov isospectral manifolds of dimension greater than 2 that have connected boundaries.

math.DG↗

arXiv2015

Changing gears: Isospectrality via eigenderivative transplantation

We introduce a new method for constructing isospectral quantum graphs that is based on transplanting derivatives of eigenfunctions. We also present simple digraphs with the same reversing zeta function, which generalizes the Bartholdi zeta function to digraphs.

math.SP↗

arXiv2015

Zeta-equivalent digraphs: Simultaneous cospectrality

We introduce a zeta function of digraphs that determines, and is determined by, the spectra of all linear combinations of the adjacency matrix, its transpose, the out-degree matrix, and the in-degree matrix. In particular, zeta-equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency, the Laplacian, the signless Laplacian, and the normalized Laplacian matrix, respectively. In addition, we express zeta-equivalence in terms of Markov chains and in terms of invasions where each edge is replaced by a fixed digraph. We finish with a method for constructing zeta-equivalent digraphs.

math.SP↗

arXiv2015

Line graphs and the transplantation method

We study isospectrality for mixed Dirichlet-Neumann boundary conditions, and extend the previously derived graph-theoretic formulation of the transplantation method. Led by the theory of Brownian motion, we introduce vertex-colored and edge-colored line graphs that give rise to block diagonal transplantation matrices. In particular, we rephrase the transplantation method in terms of representations of free semigroups, and provide a method for generating adjacency cospectral weighted directed graphs.

math.SP↗

arXiv2014

On inaudible properties of broken drums - Isospectrality with mixed Dirichlet-Neumann boundary conditions

We study isospectrality for manifolds with mixed Dirichlet-Neumann boundary conditions and express the well-known transplantation method in graph- and representation-theoretic terms. This leads to a characterization of transplantability in terms of monomial relations in finite groups and allows for the generating of new transplantable pairs from given ones as well as a computer-aided search for isospectral pairs. In particular, we show that the Dirichlet spectrum of a manifold does not determine whether it is connected and that an orbifold can be Dirichlet isospectral to a manifold.

math.DG↗

arXiv2014

Magnetic Schrödinger operators and Mañé's critical value

We study periodic magnetic Schrödinger operators on covers of closed manifolds in relation to Mañé's critical energy values of the corresponding classical Hamiltonian systems. In particular, we show that if the covering transformation group is amenable, then the bottom of the spectrum is bounded from above by Mañé's critical energy value. We also determine the spectra for various homogeneous spaces with left-invariant magnetic fields.

math.SP↗

arXiv2013

Centralizers of the infinite symmetric group

We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_\infty$. Our study is led by the double commutant relationships between finite symmetric groups and partition algebras; each approach produces a centralizer algebra that is contained in a partition algebra. Our goal is to incorporate invariants of $S_\infty$, which ties our work to the study of symmetric functions in non-commuting variables. We resultantly explore sequence spaces as permutation modules, which yields families of non-unitary representations of $S_\infty$.

math.CO↗