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

Abstract

We study isospectrality for manifolds with mixed Dirichlet-Neumann boundary conditions and express the well-known transplantation method in graph-and representationtheoretic 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.

Courant-sharp eigenvalues of Neumann 2-rep-tiles

“…By the reflection principle (see [28] for example), ϕ is an odd eigenfunction of (1.1). By Lemma 2.1 we deduce λ ∈ σ odd (D), and so there exists (p, q) ∈ O (λ) such that λ p,q = λ.…”

Line graphs and the transplantation method

“…[Kac66]. Recently, the author [Her14] Provided that ∂M is sufficiently smooth, this operator has discrete spectrum given by an unbounded non-decreasing sequence of non-negative eigenvalues.…”

“…Following [Her14], we encode each tiled manifold with mixed Dirichlet-Neumann boundary conditions by an edge-colored graph with signed loops that encode boundary conditions, e.g., G and G ′ in Figure 1(C). By definition, every vertex of an edge-colored loop-signed graph G has one incident edge of each color, either as a link to another vertex or as a signed loop.…”

“…Theorem 2. [Her14] Let G and G ′ be edge-colored loop-signed graphs with the same numbers of vertices and colors. Let M and M ′ be tiled manifolds with mixed Dirichlet-Neumann boundary conditions obtained by choosing a building block.…”

Changing gears: Isospectrality via eigenderivative transplantation

“…These graphs first appeared in [MM03] where a certain line graph construction was applied to the 7 1 pair of Dirichlet isospectral planar domains in [BCDS94]. The second author [Her15] has generalized this construction to manifolds with mixed Dirichlet-Neumann boundary conditions and revealed its connection to the graph-theoretic characterization of the famous Sunada method [Sun85] given in [Her11].…”