Fredholm theory

It should be appreciated here that the singularities at r =a and r =b makes the kernel of this equation practically separable of rank two. Fredholm theory shows that a solution to the homogeneous equation (23) requires a singular reciprocal kernel. Thus it holds that... 

The basic result of Fredholm theory dealing with equations of the second kind, the so called Fredholm Alternative, may be stated as follows for a given A, either... 

This is a Fredholm integral equation of the first kind. The regularized solution to this equation has been applied to the measurement both for the moments and the size distribution of a wide range of latices [46]. K has been given by van de Hulst [45] in terms of particle size/refractive index domain. Mie theory applies to the whole domain but in the boundary regions simpler equations have been derived. 

Referring to Eqn (7.9), we see that in any treatment of surface heterogeneity, we have to deal with three functions, any two of which, if known, assumed or determined can be used in theory to obtain the third. Equation (7.9) represents a Fredholm s integral of the first kind. The solution of equations of this type is well known to present an iU-posed or ill-conditioned problem. For our purposes, this means that the data, Q(p), can be well represented by many function pairs in the integrand hence, simply fitting the data does not guarantee that the kernel function or the distribution are individually correct. In addition, the mathematical difBculties of handling Eqn (7.9) analytically have severely restricted the number of possible variations that have been pubHshed and these are now only of historical interest. 

The theory of "impurity" or defect absorption Intensities in semiconductors has been studied by Rashba ( 1). By use of the Fredholm method, he finds that if the absorption transition occurs at k=0 and if the discrete level associated with the impurity approaches the conduction band, the intensity of the absorption line increases. The explanation offered for this intensity behavior is that the optical excitation is not localized in the impurity but encompasses a number of neighboring lattice points of the host crystal. Hence, in the absorption process, light is absorbed by the entire region of the crystal consisting of the impurity and its surroundings. 

The theory of Fredholm equations of the first kind, given by (A4.1.3), differs significantly from that of the second kind. For example, if K(x,y) has the product form (A4.1. Ip), then a solution is not possible unless fix) is a linear sum of the functions hjix). In Sect. 3.6 it was shown essentially that the equation of the second kind (3.6.7) reduces to an equation of the first kind, namely (3.6.24), with the above condition satisfied, for the case of a discrete spectrum. Pogor-zelski (1966), for example, discusses equations of the first kind in some detail. 

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