Linearized dynamic stability for vortices of Ginzburg-Landau evolutions

We consider the problem of dynamical stability for the $n$-vortex of the Ginzburg-Landau model. Vortices are one of the main examples of topological solitons, and their dynamic stability is the basic assumption of the asymptotic ``particle plus field'' description of interacting vortices. In this paper we focus on co-rotational perturbations of vortices and establish decay estimates for their linearized evolution in the relativistic case. One of the main ingredients is a construction of the distorted Fourier basis associated to the linearized operator at the vortex. The general approach follows that of Krieger-Schlag-Tataru and Krieger-Miao-Schlag and relies on the spectral analysis of Schr\"odinger operators with strongly singular potentials. Since one of the operators appearing in the linearization has zero energy solutions that oscillate at infinity, additional work is needed for our construction and to control the spectral measure. The decay estimates that we obtain are of both wave and Klein-Gordon type, and are consistent with the general theory for $2$d Schr\"odinger operators, including those that have an $s$-wave resonance, as in the present case, but faster decaying potentials. Finally, we give a new proof of the absence of unstable spectrum and provide an estimate on the location of embedded eigenvalues by using a suitable Lieb-Thirring inequality due to Ekholm and Frank. In particular, we show that eigenvalues must lie in the interval $(1.332,2)$, where $2$ represents the effective mass of one of the two scalar operators appearing in the linearization.