Half‐ and finite‐range completeness for the equation of transfer of polarized light

Abstract

On neglecting reflection by the surface the existence and uniqueness are proved for the solution of the equation of transfer of polarized light in a homogeneous semi‐infinite or finite plane‐parallel medium. A general LL‐space formulation, where 1 ≤ p < ∞, is adopted. The analysis concerns a vector‐valued convolution equation, which is an equivalent form of the equation of radiative transfer and is solved with the help of Wiener‐Hopf factorization, Fredholm index and cone preservation methods. The results are also proved for the equations obtained from the full equation of transfer by means of Fourier expansion and symmetry relations.

Reflection and transmission of polarized light: The adding method for homogeneous atmospheres

“…This problem appears to be uniquely solvable on Hp for every J+ e Q+ [Hp] (cf. [24]) and the solution at z = 0 can be expressed in terms of the Stokes vector of incident light J+(u) using a reflection operator:…”

“…On posing the problem in integral form we obtain the equivalent vector equation invertible on C(Hp)~, whereas invertibility breaks down for a= 1 (see [24]). is invertible on C(Hp)~ for as (0, 1); see [24] for the latter ingredients). We have the following convergence properties: Then (47) is immediate.…”

“…The existence and uniqueness theory for the boundary value problems (1)- (2) has been developed by van der Mee. In [24] he considered Eqs. (1)- (2) and corresponding equations for media of infinite optical thickness, while in [25] reflection by the planetary surface was incorporated.…”

Stability of Stationary Transport Equations with Accretive Collision Operators

“…The operator A must be positive semidefinite, but B=I&A need not have any compactness properties. Moreover, the solution is sought in an extension of the domain D(T) of the operator T in the original Hilbert space H. The existence and uniqueness of the solution of the stationary equation of transfer of polarized radiation were proved using invariance of the positive cone of functions having their values in the positive cone of Stokes vectors under the operators \TQ \ and B, without using that Re A is positive semidefinite [21,23]. In [24] its unique solvability was established using the accretiveness of A, with the help of the Fredholm alternative applied to the convolution integral equation version of (0.1) (0.2).…”

Polarized light transfer above a reflecting surface

“…A consistent treatment of polarized light transfer based on the (equivalent) conventions for polarization parameters of Chandrasekhar [3] and Van de Hulst 1131 is given in [12]. For notations we shall rely on this work as well as on the predecessor paper [25].…”