$s$-wave resonant short-range interactions in a $d$-dimensional finite volume

It has been known that the energy spectra of few or many particles with short-range interactions in a finite periodic box are shifted according to the size of the box. In particular, the two-body interaction in a three-dimensional box is described by the L\"{u}scher's formulas. Here we study the energy of one particle scattered by a resonant $s$-wave short-range center in a $d$-dimensional finite volume. When $d=6$, this one-body problem is mapped to the scattering of three particles in a three-dimensional box with a resonant three-body interaction.