Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem

This work deals with the interior transmission eigenvalue problem: y 00 + k 2η (r) y = 0 with boundary conditions y (0) = 0 = y 0 (1) sin k k − y (1) cos k, where the function η(r) is positive. We obtain the asymptotic distribution of non-real transmission eigenvalues under the suitable assumption on the square of the index of refraction η(r). Moreover, we provide a uniqueness theorem for the case R 1 0 p η(r)dr > 1, by using all transmission eigenvalues (including their multiplicities) along with a partial information of η(r) on the subinterval. The relationship between the proportion of the needed transmission eigenvalues and the length of the subinterval on the given η(r) is also obtained