= Frequency-regulated Dirichlet Casimir energy
{title2=$E_{\mathrm{Cas}}(L)=-\pi\hbar c/(24L)$}
For a massless real scalar in one spatial dimension, applying $e^{-\omega/(\pi\Lambda)}$ to both the Dirichlet mode sum and the unbounded-line integral gives the same leading bulk energy $\pi L\Lambda^2/2$. Their finite difference is $-\pi/(24L)$ in units $\hbar=c=1$, producing the attractive force $-\pi/(24L^2)$. This scalar boundary force is distinct from the retarded atom-surface Casimir-Polder interaction.
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