  This paper is devoted to the discreteness of the transmission eigenvalue
problems. It is known that this problem is not self-adjoint and a priori
estimates are non-standard and do not hold in general. Two approaches are used.
The first one is based on the multiplier technique and the second one is based
on the Fourier analysis. The key point of the analysis is to establish the
compactness and the uniqueness for Cauchy problems under various conditions.
Using these approaches, we are able to rediscover quite a few known
discreteness results in the literature and obtain various new results for which
only the information near the boundary are required and there might be no
contrast of the coefficients on the boundary.
