Boundary value problems and Heisenberg uniqueness pairs

We describe a general method for constructing Heisenberg uniqueness pairs \((\Gamma ,\Lambda )\) $(\mathrm{\Gamma },\mathrm{\Lambda })$ in the euclidean space \(\mathbb {R}^{n}\) ${\mathbb{R}}^n$ based on the study of boundary value problems for partial differential equations. As a result, we show, for instance, that any pair made of the boundary \(\Gamma \) $\mathrm{\Gamma }$ of a bounded convex set \(\Omega \) $\mathrm{\Omega }$ and a sphere \(\Lambda \) $\mathrm{\Lambda }$ is an Heisenberg uniqueness pair if and only if the square of the radius of \(\Lambda \) $\mathrm{\Lambda }$ is not an eigenvalue of the Laplacian on \(\Omega \) $\mathrm{\Omega }$ . The main ingredients for the proofs are the Paley–Wiener theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial boundary value problem, the continuity of single layer potentials, and some complex analysis in \(\mathbb {C}^{n}\) ${\mathbb{C}}^n$ . Denjoy’s theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.