The curious case of operators with spectral density increasing as $\Omega(E)\sim e^{\,\mathrm{Const.}\, E^2}$

Motivated by a putative model of black holes as quantum objects we consider what types of operators would have a corresponding spectrum. We find that there are indeed such operators, but of a rather unusual types, and for which the wave functions are only barely localized. We point out a tension between such almost delocalized states and black holes as compact objects.