Asymptotic linear stability of columnar vortices driven by Coriolis force

In this paper, we establish the asymptotic linear stability of a class of Coriolis-driven columnar vortices for the 3-D axisymmetric Euler equations. This result represents a critical step toward proving the nonlinear asymptotic stability of such vortices. The key and widely applicable strategy is to construct a distorted Fourier basis, which is achieved by solving a two-parameter $(c, \xi)$-dependent Schr\"odinger equation associated with the linearized operator of the system. To capture the precise asymptotic behavior of the solution, we decompose the $c-\xi$ plane into distinct regions, with the partitioning guided by the leading-order profiles of the Schr\"odinger equation across different parameter regimes.