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

Abstract

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 $b_L\ne -b_R$. 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.