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

Abstract

'Alexander Tovbis'] In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $\mathcal{H}_L:L^2[b_L,0]\rightarrow L^2[0,b_R]$ and $\mathcal{H}_R:L^2[0,b_R]\rightarrow L^2[b_L,0]$. 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.