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 energy levels in a large box are approximately described by the L\"{u}scher's formula. 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 closely related to the scattering of three particles in a three-dimensional box with a resonant three-body interaction. For $d\ge4$, we derive systematic expansions of the low-lying energy eigenvalues at large box sizes.