Entropy of very low energy localized states

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.