The Pompeiu and Schiffer conjectures are longstanding problems in spectral geometry asking whether the disk is the only bounded simply connected domain for which there exists a nonzero function whose integral over every rigid motion of the domain vanishes, or equivalently, whether it is the only domain admitting a nonconstant Neumann eigenfunction that is constant on the boundary. Here we present a computer-assisted construction of a noncircular real-analytic counterexample disproving both conjectures.

Preprint:

https://arxiv.org/pdf/2608.01579