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In this paper we extend Sylvester's approach via upper triangular compactoperators to establish the discreteness of transmission eigenvalues forhigher-order main terms and higher-order perturbations. The coefficients of theperturbations must be sufficiently smooth and the coefficients of thehigher-order terms of the perturbation must vanish in a neighbourhood of theboundary of the underlying domain. The zeroeth order term must satisfy asuitable coercivity condition in a neighbourhood of the boundary.

Discreteness of Transmission Eigenvalues for Higher-Order Main Terms and Perturbations