Empirical plunge profiles of time-frequency localization operators

For time-frequency localization operators, related to the short-time Fourier transform, with symbol $R\mathrm{\Omega }$, we work out the exact large $R$ eigenvalue behavior for rotationally invariant $\mathrm{\Omega }$ and conjecture that the same relation holds for all scaled symbols $R\mathrm{\Omega }$ as long as the window is the standard Gaussian. Specifically, we conjecture that the $k$-th eigenvalue of the localization operator with symbol $R\mathrm{\Omega }$ converges to $\frac{1}{2}\mathrm{erfc}\left(\sqrt{2\pi }\frac{k-R^2\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{R\mathrm{\mid }\mathrm{\partial }\mathrm{\Omega }\mathrm{\mid }}\right)$ as $R\to \mathrm{\infty }$. To support the conjecture, we compute the eigenvalues of discrete frame multipliers with various symbols using LTFAT and find that they agree with the behavior of the conjecture to a large degree.