Erik Aurell: Operators with spectral density increasing as Omega(E) ~ exp[cE^2]

Abstract: The quantum nature of black holes has remained a key problem in fundamental physics since the discovery of Hawking radiation, now more than half a century ago. It was early on pointed out by Bekenstein and Mukhanov that if black holes are quantum objects with a density of states corresponding to Bekenstein–Hawking entropy, then the density of states of those objects must grow as \(e^{Const. E^2}\). An entropy growing quadratically with is a concave and not a convex function, and the canonical distribution is not defined for any finite temperature (the partition function is infinite). This already says that a quantum black hole, if such a thing exists, must be a thing with rather exotic properties.

While natural objects with sub-exponential or exponential spectral growth, i.e. \(e^{Const. E^a}\) with \(a\leq 1\) are known in both physics and
mathematics, if mathematicians also know of natural objects with super-exponential spectral growth, that knowledge has not reached outsiders.

I will present a simple physical toy model which does have super-exponential growth, though it obviously cannot be a reasonable model of a quantum black hole as all states are very delocalized. I will then use this observation as starting point of a discussion of why it is hard to construct models of compact objects with super-exponential spectral growth, and what this may perhaps tell us about quantum black holes.

The talk is based on joint work with Satya N. Majumdar Physical Review Research 7: 043165 (2025) [arXiv:2504.06623].