Novel black hole bound states and entropy

Details Novel black hole bound states and entropy Novel black hole bound states and entropy

2011-02-24

arxiv.org/abs/1102.4919v3

Phys. Rev. D 83, 124045 (2011)

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

13 pages, 3 figures, approximate formula for energy spectrum added at the end of section 2.1 along with additional minor chang

Scientific paper

We solve for the spectrum of the Laplacian as a Hamiltonian on $\mathbb{R}^{2}-\mathbb{D}$ and in $\mathbb{R}^{3}-\mathbb{B}$. A self-adjointness analysis with $\partial\mathbb{D}$ and $\partial\mathbb{B}$ as the boundary for the two cases shows that a general class of boundary conditions for which the Hamiltonian operator is essentially self-adjoint are of the mixed (Robin) type. With this class of boundary conditions we obtain "bound state" solutions for the Schroedinger equation. Interestingly, these solutions are all localized near the boundary. We further show that the number of bound states is finite and is in fact proportional to the perimeter or area of the removed \emph{disc} or \emph{ball}. We then argue that similar considerations should hold for static black hole backgrounds with the horizon treated as the boundary.

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