Inverse problems for the zeros of the Wigner function

Abstract

We study the nodal (zero) set of the Wigner transform, obtaining a rigidity theorem for the eigenstates of one-dimensional harmonic oscillator. For every $k \in \mathbb{N}_0$, and $f\in L^{2}(\mathbb{R})$, equality between the nodal sets of $Wf$ and $Wh_{k}$ determines $f$, up to a constant phase and an explicitly defined nodal-preserving pseudo-displacement. The proof requires a result with independent interest, where it is shown that a Wigner distribution has bounded nodal set if and only if its underlying state is, up to time-frequency and metaplectic symmetries, a finite Hermite expansion. The demonstration of the results combines new analytic, geometric and arithmetic tools: a property of classical convolution, a geometric study of phase space displacements and a new divisibility theorem for generalized Laguerre polynomials. As a by-product, we obtain several results concerning the structure and admissible geometry of bounded nodal sets of Wigner distributions.