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A solution method for the linear Chandrasekhar equation. (English) Zbl 1110.45002
The authors consider the linearized Chandrasekhar-Ambarzumyan equation: \[ \begin{aligned} A\varphi(x):= & \varphi(x)-x\int_0^1\frac{\varphi(x)k(s) +k(x)\varphi(s)}{s+x}\,ds\\ = & c(x)\varphi(x)-xk(x)\int_0^1\frac{\varphi(s)\,ds}{s +x}=g(x),\qquad 0<x<1\, .\end{aligned} \] Due to the multiplier \(x\) in front of the integral with fixed singularity, \(A=cI+T\), where \(T\) is a compact operator and the coefficient is peculiar: \[ c(x):=1-x\int_0^1\frac{k(s)}{s+x}\,ds. \] A Stiltjes-type transform reduces the equation to a boundary value problem for harmonic functions \(\Phi(x)\) in the upper half-plane, solved in closed form. Under additional constraints a solution \(\Phi(x)\) extends holomorphically in the lower half-plane slit along a straight line segment and the solution of the original problem is recovered as the values of \(\Phi(x)\) on the lips of this slit. The authors present explicit criteria for the Fredholm property of \(A\) and formulae for the complete set of solutions.

MSC:
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
30E25 Boundary value problems in the complex plane
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