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We introduce an analog of the Dirichlet-to-Neumann map for the Maxwellequation in a bounded domain. We show that it can be approximated by apseudodifferential operator on the boundary with a matrix-valued symbol and wecompute the principal symbol. As a consequence, we obtain a parabolic regionfree of the transmission eigenvalues associated to the Maxwell equation.

Semiclassical parametrix for the Maxwell equation and applications to the electromagnetic transmission eigenvalues