  We show that solutions to Smoluchowski's equation with a constant coagulation
kernel and an initial datum with some regularity and exponentially decaying
tail converge exponentially fast to a self-similar profile. This convergence
holds in a weighted Sobolev norm which implies the L\^2 convergence of
derivatives up to a certain order k depending on the regularity of the initial
condition. We prove these results through the study of the linearized
coagulation equation in self-similar variables, for which we show a spectral
gap in a scale of weighted Sobolev spaces. We also take advantage of the fact
that the Laplace or Fourier transforms of this equation can be explicitly
solved in this case.
