Sharp estimates for eigenvalues of localization operators with applications to area laws

We study the eigenvalues of the localization operator $S_{A,B}=P_A{\mathcal{F}}^{-1}{P}_B\mathcal{F}{P}_A$, where $\mathcal{F}$ is the Fourier transform and $A=c{A}_0,B=B_0$ for some fixed sets $A_0,B_0\subset {\mathbb{R}}^d$ and a large parameter $c>0$. For the counting function of the eigenvalues $\mathrm{\mid }\{n:\epsilon <{\lambda }_n(A,B)\le 1-\epsilon \}\mathrm{\mid }$ we obtain a sharp uniform upper bound if one of the sets is a finite disjoint union of parallelepipeds and a bound which is only a single logarithm off the conjectural optimal bound in the general case. These bounds are applied to the estimation of traces $\mathrm{T}\mathrm{r}f(S_{A,B})$ for functions $f$ with a very low regularity, in particular establishing an enhanced area law in the former case.