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 $$\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])$$ H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and $$\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])$$ 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 $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L has been previously obtained, but only asymptotically when $$b_L\ne -b_R$$ b L ≠ - b R . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.