On spectral analysis of an integral-difference operator (Q1375379)

In the Hilbert space \(L_2 (\mathbb{R})\) the author considers the following integral operator \[ (K_{\chi_a} u)(x)= \int^\infty_{-\infty} {\chi_a (s)u(x) -\chi_a(x)u(s) \over |x-s|} ds, \] where \(\chi_a(x)\) is the indicator of the interval \([-a,a]\); such an operator arises as a collision operator in statistical physics models. He gives a complete spectral description for the operator \(K_{\chi_a}\) and for the restricted operator \(\chi_a K_{\chi_a} \chi_a\) acting in the space \(L_2 (-a,a)\).