  We expand on previous work involving "vacuum-bounded" states, i.e., states
such that every measurement performed outside a specified interior region gives
the same result as in the vacuum. We improve our previous techniques by
removing the need for a finite outside region in numerical calculations. We
apply these techniques to the limit of very low energies and show that the
entropy of a vacuum-bounded state can be much higher than that of a rigid box
state with the same energy. For a fixed $E$ we let $L_in'$ be the length of a
rigid box which gives the same entropy as a vacuum-bounded state of length
$L_in$. In the $E\to 0$ limit we conjecture that the ratio $L_in'/L_in$ grows
without bound and support this conjecture with numerical computations.
