Distribution of transmission eigenvalues and inverse spectral analysis with partial information on the refractive index

In this work, we consider the interior transmission eigenvalue problem for a spherically stratified medium, which can be formulated as y ′′ ( r ) + k 2 η ( r ) y ( r ) = 0 endowed with boundary conditions y ( 0 ) = 0 = y ′ ( 1 ) sin k k − y ( 1 ) cos k , where the refractive index η ( r ) is positive and real. We obtain the distribution of transmission eigenvalues under assumptions that a : = ∫ 0 1η ( r ) dr = 1 and one of these conditions that (i) η (1) ≠ 1, (ii) η (1) = 1, η ′(1) ≠ 0, and (iii) η (1) = 1, η ′(1) = 0, η ″ (1) ≠ 0, respectively. Moreover, in the case a = 1, we prove that if partial information on η ( r ) is known on subdomain, then only a part of eigenvalues can uniquely determine η ( r ) on the whole interval, and the relationship between the proportion of the missing eigenvalues and the subinterval of the known information on η ( r ) is revealed. Copyright © 2016 John Wiley & Sons, Ltd.