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

Novel black hole bound states and entropy