Boundary conditions and the entropy bound

The entropy-to-energy bound is examined for a quantum scalar field confined to a cavity and satisfying Robin’s condition on the boundary of the cavity. It is found that near certain points in the space of the parameter defining the boundary condition the lowest eigenfrequency (while nonzero) becomes arbitrarily small. Estimating, following Bekenstein and Schiffer, the ratio $S/E$ by the $\zeta$-function $\left(24\zeta \mathrm{}\right(4)\mathrm{}{)}^{1/4},$ we compute $\zeta \mathrm{}\left(4\right)\mathrm{}$ explicitly and find that it is not bounded near those points that signal violation of the bound. We interpret our results as imposing certain constraints on the value of the boundary interaction and estimate the forbidden region in the parameter space of the boundary conditions.