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

TL;DR

This paper investigates isospectrality of manifolds with mixed boundary conditions, using group theory to characterize transplantability, revealing that Dirichlet spectra do not determine connectivity and that orbifolds can be isospectral to manifolds.

Contribution

It introduces a group-theoretic framework for analyzing transplantability and generates new isospectral pairs, expanding understanding of spectral properties under mixed boundary conditions.

Findings

Dirichlet spectrum does not determine connectivity

Orbifolds can be Dirichlet isospectral to manifolds

New transplantable pairs can be generated computationally

Abstract

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.