1 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec H3G 1M8 Canada.
2 SISSA, International School for Advanced Studies, Via Bonomea 265, Trieste, Italy.
3 Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 114 28 Stockholm, Sweden.
4 Department of Mathematics, University of Central Florida, P.O. Box 161364, 4000 Central Florida Blvd, Orlando, FL 32816-1364 USA.

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