Quasi-Normal Modes: The “Electrons” of Black Holes as “Gravitational Atoms”? Implications for the Black Hole Information Puzzle

Some recent important results on black hole (BH) quantum physics concerning theBH effective stateand the natural correspondence between Hawking radiation and BH quasi-normal modes (QNMs) are reviewed, clarified, and refined. Such a correspondence permits one to naturally interpret QNMs as quantum levels in a semiclassical model. This is a model of BH somewhat similar to the historical semiclassical model of the structure of a hydrogen atom introduced by Bohr in 1913. In a certain sense, QNMs represent the “electron” which jumps from a level to another one and the absolute values of the QNMs frequencies, “triggered” by emissions (Hawking radiation) and absorption of particles, represent the energy “shells” of the “gravitational hydrogen atom.” Important consequences on the BH information puzzle are discussed. In fact, it is shown that the time evolution of this “Bohr-like BH model” obeysa time dependent Schrödinger equationwhich permits the final BH state to bea purequantum state instead of a mixed one. Thus, information comes out in BH evaporation in agreement with the assumption by ’t Hooft that Schröedinger equations can be used universally for all dynamics in the universe. We also show that, in addition, our approach solves the entanglement problem connected with the information paradox.

Exceptional Points for Nonlinear Schroedinger Equations Describing Bose-Einstein Condensates of Ultracold Atomic Gases

The coalescence of two eigenfunctions with the same energy eigenvalue is not possible in Hermitian Hamiltonians. It is, however, a phenomenon well known from non-hermitian quantum mechanics. It can appear, e.g., for resonances in open systems, with complex energy eigenvalues. If two eigenvalues of a quantum mechanical system which depends on two or more parameters pass through such a branch point singularity at a critical set of parameters, the point in the parameter space is called an exceptional point. We will demonstrate that exceptional points occur not only for non-hermitean Hamiltonians but also in the nonlinear Schroedinger equations which describe Bose-Einstein condensates, i.e., the Gross-Pitaevskii equation for condensates with a short-range contact interaction, and with additional long-range interactions. Typically, in these condensates the exceptional points are also found to be bifurcation points in parameter space. For condensates with a gravity-like interaction between the atoms, these findings can be confirmed in an analytical way.