The distorted Fourier transform for the linearized Gross-Pitaevskii equation in the Hyperbolic plane

Motivated by the stability problem for Ginzburg-Landau vortices on the hyperbolic plane, we develop the distorted Fourier transform for a general class of radial non-self-adjoint matrix Schrödinger operators on the hyperbolic plane. This applies in particular to the operator obtained by linearizing the equivariant Ginzburg-Landau equation on the hyperbolic plane around the degree one vortex. We systematically construct the distorted Fourier transform by writing the Stone formula for complex energies and taking the limit as the energy tends to the spectrum of the operator on the real line. This approach entails a careful analysis of the resolvent for complex energies in a neighborhood of the real line. It is the analogue of the approaches used in \cite{KS,ES2, LSS25}, where the limiting operator as $r\to\infty$ is not self-adjoint and which we carry out for all energies. Our analysis serves as the starting point for the study of the stability of the Ginzburg-Landau vortex under equivariant perturbations.