The Riemann–Hilbert Approach to the Inversion of

Abstract

In the present chapter we focus on a class of singular integral equation driven by a one parameter $\gamma$-regularisation of the operator $\mathcal {S}_N$. More precisely, we introduce the singular integral operator $\mathcal {S}_{N;\gamma }$

(4.0.1)

This operator is a regularisation of the operator $\mathcal {S}_N$ in the sense that, formally, $\mathcal {S}_{N;\infty }=\mathcal {S}_{N}$. This regularisation enables to set a well defined associated Riemann–Hilbert problem, and is such that, once all calculations have been done and the inverse of $\mathcal {S}_{N;\gamma }$ constructed, we can send $\gamma \rightarrow +\infty$ at the level of the obtained answer. It is then not a problem to check that this limiting operator does indeed provide one with the inverse of $\mathcal {S}_N$. We start this analysis by, first, recasting the singular integral equation into a form where the variables have been re-scaled. Then, we put the problem of inverting the re-scaled operator associated with $\mathcal {S}_{N;\gamma }$ in correspondence with a vector valued Riemann-Hilbert problem. The resolution of this vector problem demands the resolution of a $2\times 2$ matrix Riemann–Hilbert problem for an auxiliary matrix $\chi$. We construct the solution to this problem, for N-large enough, in Section 4.2.2 and then exhibit some of the overall properties of the solution $\chi$ in Section 4.2.3. We shall build on these results so as to invert $\mathcal {S}_{N;\gamma }$ and then $\mathcal {S}_N$ in subsequent sections.