= Dirichlet scalar vacuum energy density
{c}
{title2=$\langle\mathcal H(x)\rangle=\sum_n[\omega_n/(2L)-m^2\cos(2k_nx)/(2\omega_nL)]$}
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.
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