Exact solution of a confined two-particle system with calculation of its dead-layer size

A molecule consisting of two particles interacting with a delta-function attraction, and confined to a one-dimensional volume, is studied. From the exact solution of the system we deduce that the centre-of-mass effectively moves in a volume reduced by an inaccessible dead layer near each wall. For a very large volume the dead-layer width equals the size of the molecule, defined as the expectation value of the interparticle distance for the unconfined molecule. For finite volumes the dead-layer width is smaller than this. For unequal particle masses, perturbation in the mass difference up to third order yields the same result, and we conjecture that for this system the dead-layer width is independent of the mass values.