Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms [Formula: see text] and [Formula: see text] . These operators arise when one studies the interior problem of tomography. The diagonalization of [Formula: see text] has been previously obtained, but only asymptotically when [Formula: see text] . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes [Formula: see text] explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.