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

TL;DR

This paper introduces a novel Riemann-Hilbert problem approach to explicitly diagonalize finite Hilbert transform operators on adjacent intervals, providing detailed spectral analysis and asymptotic behavior.

Contribution

It develops a new method using matrix Riemann-Hilbert problems to explicitly diagonalize finite Hilbert transforms, improving upon previous asymptotic results.

Findings

Explicit diagonalization of $\\mathcal{H}_R, \mathcal{H}_L$ operators

Asymptotic analysis of related RHP solutions

Error estimates for the diagonalization

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 $b_L\neq-b_R$. 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.