  Extending work of Carty, we show that $H^1$ solutions of a simplified 1D BGK
model decay exponentially in $L^2$ to a subclass of the class of grossly
determined solutions as defined by Truesdell and Muncaster. In the process, we
determine the spectrum and generalized eigenfunctions of the associated
non-selfadjoint linearized operator and derive the associated generalized
Fourier transform and Parseval's identity. Notably, our analysis makes use of
rigged space techniques originating from quantum mechanics, as adapted by
Ljance and others to the nonselfadjoint case.
