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We study the point spectrum of the linearisation of Euler's equation for theideal fluid on the torus about a shear flow. By separation of variables theproblem is reduced to the spectral theory of a complex Hill's equation. UsingHill's determinant an Evans function of the original Euler equation isconstructed. The Evans function allows us to completely characterise the pointspectrum of the linearisation, and to count the isolated eigenvalues withnon-zero real part. We prove that the number of discrete eigenvalues of helinearised operator for a specific shear flow is exactly twice the number ofnon-zero integer lattice points inside the so-called unstable disk.

An Evans function for the linearised 2D Euler equations using Hill's determinant