Mathematics
Scientific paper
Aug 1983
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1983jqsrt..30..119d&link_type=abstract
Journal of Quantitative Spectroscopy and Radiative Transfer (ISSN 0022-4073), vol. 30, Aug. 1983, p. 119-129.
Mathematics
8
Orthogonal Functions, Polarized Light, Radiative Transfer, Transfer Functions, Eigenvalues, Eigenvectors, Green'S Functions, Transport Theory
Scientific paper
It is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation. The adjoints are obtained in terms of Case's eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green's functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green's functions and their adjoints. Linear and nonlinear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.
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