Spectrum of three-body bound states in a finite volume

The spectrum of a bound state of three identical particles with a mass $m$ in a finite cubic box is studied. It is shown that in the unitary limit, the energy shift of a shallow bound state is given by $\Delta E=c (\kappa^2/m),(\kappa L)^{-3/2}|A|^2\exp(-2\kappa L/\sqrt{3})$,where$\kappa$isthebound-statemomentum,$L$istheboxsize,$|A|^2$ denotes the three-body analog of the asymptotic normalization coefficient of the bound state wave function and $c$ is a numerical constant. The formula is valid for $\kappa L\gg 1$.