A note on the Chandrasekhar's X-function (Q1105806)

Consider the Volterra integral equation \(J(x,z)=\exp (ixz)+\int^{t}_{0}K(x-y)J(y,z)dy\) Chandrasekhar's X and Y functions are the limits defined by \(X(z)=\lim_{x\to 0^+}J(x,z)\) and \(Y(z)=\lim_{x\to t^-}J(x,z)\). Generalizing a result of \textit{R. V. Rao} [J. Math. Analysis Appl. 59, 60-68 (1977; Zbl 0353.45004)], the author establishes two theorems. The first one follows Theorem 1: The unique solution \(\alpha(x,t)\) of \(\alpha (x,t)=K(x)+\int^{t}_{0}K(x- y)\alpha (y,t)dy\) is differentiable with respect to t for almost every t and its derivative is \(\alpha (t,t)\cdot \alpha_ 1(x-t,t)\) in the following sense. \(\lim_{h\to 0} \int^{t}_{0} | ((\alpha(x,t+h)- \alpha(x,t))/h)- \alpha(t,t)\alpha_ 1(x-t,t)| dx=0\) where \(\alpha_ 1(x,t)\) is the unique solution of the integral equation \(\alpha_ 1(x,t)=K(x)+\int^{0}_{-t}K(x-y)\alpha_ 1(y,t)dy\). The theorems are claimed to be useful in obtaining a relation between the eigenvalues and the X-function for normal operators. [Some printing errors can be detected by a reader.]