One can't hear orientability of surfaces

TL;DR

This paper demonstrates that the orientability of a surface with boundary cannot be determined solely from its Neumann spectrum by constructing isospectral pairs with different orientability.

Contribution

It provides the first explicit construction of isospectral surfaces with boundary that differ in orientability, using Sunada's and Buser's methods within orbifold frameworks.

Findings

Constructed isospectral orientable and non-orientable surfaces with boundary.

Proved these surfaces have identical Neumann spectra.

Showed they have different Dirichlet spectra.

Abstract

The main result of this paper is that one cannot hear orientability of a surface with boundary. More precisely, we construct two isospectral flat surfaces with boundary with the same Neumann spectrum, one orientable, the other non-orientable. For this purpose, we apply Sunada's and Buser's methods in the framework of orbifolds. Choosing a symmetric tile in our construction, and adapting a folklore argument of Fefferman, we also show that the surfaces have different Dirichlet spectra. These results were announced in the {\it C. R. Acad. Sci. Paris S\'er. I Math.}, volume 320 in 1995, but the full proofs so far have only circulated in preprint form.