  We analyse an operator arising in the description of singular solutions to
the two-dimensional Keller-Segel problem. It corresponds to the linearised
operator in parabolic self-similar variables, close to a concentrated
stationary state. This is a two-scale problem, with a vanishing thin transition
zone near the origin. Via rigorous matched asymptotic expansions, we describe
the eigenvalues and eigenfunctions precisely. We also show a stability result
with respect to suitable perturbations, as well as a coercivity estimate for
the non-radial part. These results are used as key arguments in a new rigorous
proof of the existence and refined description of singular solutions for the
Keller-Segel problem by the authors. The present paper extends the result by
Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [Physica D, 2012]. Two major
difficulties arise in the analysis: this is a singular limit problem, and a
degeneracy causes corrections not being polynomial but logarithmic with respect
to the main parameter.
