A Quantum Perspective on Uniqueness Pairs for the Fourier Transform

Abstract: Kulikov, Nazarov, and Sodin recently introduced the notion of criticality to analyze when discrete subsets of the real line form uniqueness pairs for the Fourier transform, relying crucially on estimates derived from the Wirtinger-Poincar\'{e} inequality. In this work, drawing analogies from quantum mechanics, we propose an alternative approach to studying uniqueness sets in a Fourier-symmetric Sobolev space. Our method is based on eigenvalue estimates and offers a simplified perspective on the problem.