  We investigate solutions of the 2d incompressible Euler equations, linearized
around steady states which are radially decreasing vortices. Our main goal is
to understand the smoothness of what we call the spectral density function
associated with the linearized operator, which we hope will be a step towards
proving full nonlinear asymptotic stability of radially decreasing vortices.
The motivation for considering the spectral density function is that it is not
possible to describe the vorticity or the stream function in terms of one
modulated profile. There are in fact two profiles, both at the level of the
physical vorticity and at the level of the stream function. The spectral
density function allows us to identify these profiles, and its smoothness leads
to pointwise decay of the stream function which is consistent with the decay
estimates first proved in Bedrossian-Coti Zelati-Vicol in [5].
