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We prove trace class estimates for self-adjoint elliptic operators defined in $\R[n]$ or $\R[]_+$. These estimates are also applicable when a physical system is governed by a wave equation by employing separation of variables to obtain corresponding temporal and spatial Hamiltonians. It is shown, in one important example, that the resulting Hamiltonians are of trace class such that quantum statistics can be applied to the system.

trace class estimates, Hilbert-Schmidt Sobolev embeddings, quantum statistics, quantization of gravity, black hole, gravitational wave, wave equation, partition function