Integrating the Wigner Distribution on Subsets of the Phase Space, a Survey
Résumé
We review several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, we provide a theoretical proof of the invalidity of Flandrin's conjecture, a fact already proven via numerical arguments in our joint paper [J. Fourier Anal. Appl. \textbf{26} (2020), no.~1, paper 6] with B.~Delourme and T.~Duyckaerts. We use also the J.G.~Wood \& A.J.~Bracken paper [J. Math. Phys. \textbf{46} (2005), no.~4, 042103, 14], for which we offer a mathematical perspective. We review thoroughly the case of subsets of the plane whose boundary is a conic curve and show that Mehler's formula can be helpful in the analysis of these cases, including for the higher dimensional case investigated in the paper [J. Math. Phys. \textbf{51} (2010), no.~10, 102101, 6] by E.~Lieb and Y.~Ostrover. Using the Feichtinger algebra, we show that, generically in the Baire sense, the Wigner distribution of a pulse in $L^2(\mathbb R^n)$ does not belong to $L^1(\mathbb R^{2n})$, providing as a byproduct a large class of examples of subsets of the phase space $\mathbb R^{2n}$ on which the integral of the Wigner distribution is infinite. We study as well the case of convex polygons of the plane, with a rather weak estimate depending on the number of vertices, but independent of the area of the polygon.