T -matrix approach to the phonon-mediated Casimir interaction

We develop a theory of the phonon-mediated Casimir interaction between two pointlike impurities which is based on the single-impurity scattering $T$-matrix approach. We show that the Casimir interaction at $T=0$ falls off as a power law with the distance between the impurities. We find that the power in the weak and unitary phonon-impurity scattering limits differs, and we relate the power law to the low-energy properties of the single-impurity scattering $T$ matrix. In addition, we consider the Casimir interaction at finite temperature and show that at finite temperatures the Casimir interaction becomes exponential at large distances.