  In $D=4,N=2$ theories on $R^{3,1}$, the index receives contributions not only
from single-particle BPS states, counted by the BPS indices, but also from
multi-particle states made of BPS constituents. In a recent work
[arXiv:1406.2360], a general formula expressing the index in terms of the BPS
indices was proposed, which is smooth across walls of marginal stability and
reproduces the expected single-particle contributions. In this note, I analyze
the two-particle contributions predicted by this formula, and show agreement
with the spectral asymmetry of the continuum of scattering states in the
supersymmetric quantum mechanics of two non-relativistic, mutually non-local
dyons. This provides a physical justification for the error function profile
used in the mathematics literature on indefinite theta series, and in the
physics literature on black hole partition functions.
