Eigenvalues of the linearized 2D Euler equations via Birman-Schwinger and Lin's operators

We study spectral instability of steady states to the linearized 2D Euler equations on the torus written in vorticity form via certain Birman-Schwinger type operators $K_{\lambda}(\mu)$ and their associated 2-modified perturbation determinants $\mathcal D(\lambda,\mu)$. Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator $L_{\rm vor}$ in terms of zeros of the 2-modified Fredholm determinant $\mathcal D(\lambda,0)=\det_{2}(I-K_{\lambda}(0))$ associated with the Hilbert Schmidt operator $K_{\lambda}(\mu)$ for $\mu=0$. As a consequence, we are also able to provide an alternative proof to an instability theorem first proved by Zhiwu Lin which relates existence of an unstable eigenvalue for $L_{\rm vor}$ to the number of negative eigenvalues of a limiting elliptic dispersion operator $A_{0}$.