Dirichlet scalar vacuum energy density ($⟨\mathcal{H}(x)⟩={\sum }_n[{\omega }_n/(2L)-m^2\cos (2k_{n}x)/(2{\omega }_{n}L)]$)

On a one-dimensional interval with Dirichlet boundary conditions, $k_n=n\pi /L$ and ${\omega }_n^2=k_n^2+m^2$. The massive-field vacuum energy density can depend on position even though its integral is the oscillator sum ${\sum }_n{\omega }_n/2$. All terms require a common ultraviolet regulator when vacuum energies in different geometries are compared.