  This paper introduces a numerical method for computing the spectrum of
adjoint (left) eigenfunctions of spiral wave solutions to reaction-diffusion
systems in arbitrary geometries. The method is illustrated by computing over a
hundred eigenfunctions associated with an unstable time-periodic single-spiral
solution of the Karma model on a square domain. We show that all leading
adjoint eigenfunctions are exponentially localized in the vicinity of the
spiral tip, although the marginal modes (response functions) demonstrate the
strongest localization. We also discuss the implications of the localization
for the dynamics and control of unstable spiral waves. In particular, the
interaction with no-flux boundaries leads to a drift of spiral waves which can
be understood with the help of the response functions.
