Eigenfunctions, free boundaries, and time-frequency localization

We develop an inverse theory for time-frequency localization operators, whose central idea that of is free-boundary problem: the localization domain is unknown and its boundary is recovered from prescribed spectral data. The approach is based on the principle that an eigenfunction may be regarded as geometric data which determines a localization domain, and prescribing it has strong consequences for the associated variational problem.
Four main results follow from this main framework. First, if $f_0$ is a polynomial sufficiently close to the Gaussian and $\lambda\in(0,1)$, we construct a real-analytic domain $U_\lambda$ such that $f_0$ is an eigenfunction of the localization operator associated with $U_\lambda$ and with eigenvalue $\lambda$, giving a general inverse construction of localization domains, which is the first in the literature. Second, we recover the null-set invariant Abreu-Dörfler characterization of disks as the only localization domains of Hermite polynomials in the simply connected case. Third, we prove optimality of the exponent $1/2$ in the Gómez-Guerra-Ramos-Tilli quantitative stability inequality, answering thus a question posed by those authors. Finally, we show that local maximizers of the Gaussian Faber-Krahn problem are disks, which extends the Nicola-Tilli concentration inequality to the local case as well, and, as a matter of fact, as a consequence of a Fock space concentration-compactness profile decomposition, we are able to use this to give a new, different proof of the Nicola-Tilli theorem.