  In this paper we study the spectra of bounded self-adjoint linear operators
that are related to finite Hilbert transforms $\mathcal{H}_L:L^2([b_L,0])\to
L^2([0,b_R])$ and $\mathcal{H}_R:L^2([0,b_R])\to L^2([b_L,0])$. These operators
arise when one studies the interior problem of tomography. The diagonalization
of $\mathcal{H}_R,\mathcal{H}_L$ has been previously obtained, but only
asymptotically when $b_L\neq-b_R$. We implement a novel approach based on the
method of matrix Riemann-Hilbert problems (RHP) which diagonalizes
$\mathcal{H}_R,\mathcal{H}_L$ explicitly. We also find the asymptotics of the
solution to a related RHP and obtain error estimates.
