Volume Dependence of the Energy of an Enclosed Quantum Mechanical System

A sum rule is developed that allows one to write a closed form expression for $\frac{d^2{E}_n}{d{V}^2}$; where $E_n\mathrm{}\left(V\right)\mathrm{}$ is the $n\mathrm{th}$ eigenvalue and $V$ the volume of an enclosed many-body system with arbitrary interaction potential. This results in a simple differential equation for $E_n\mathrm{}\left(V\right)\mathrm{}$. Unfortunately, the solution, $E_n\mathrm{}\left(V\right)\mathrm{}$, does not appear to be physically meaningful. The reason for this is the appearance of unacceptable functions in the formal development of the sum rule. Because these functions do not belong to the linear manifold with respect to which the Hamiltonian is Hermitian, considerable care must be exercised in applying reduction formulas. Several examples are considered in detail.