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Distribution of transmission eigenvalues and inverse spectral analysis with partial information on the refractive index
Author(s) -
Xu XiaoChuan,
Xu XinJian,
Yang ChuanFu
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3918
Subject(s) - eigenvalues and eigenvectors , mathematics , inverse , refractive index , transmission (telecommunications) , distribution (mathematics) , interval (graph theory) , mathematical analysis , boundary values , boundary (topology) , index (typography) , combinatorics , boundary value problem , geometry , optics , physics , telecommunications , quantum mechanics , world wide web , computer science
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.