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 H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and H R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . 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 ≠ - 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.