Riemann-Hilbert approach to Slavnov overlaps in the Lieb-Liniger model

We provide a method to compute Slavnov overlaps in the Lieb-Liniger model using the steepest descent method of the Riemann-Hilbert problem. To do so, we employ the Matsuo-Kostov representation of the Slavnov overlaps to derive an integral equation for the respective resolvent, and then represent this equation as a Riemann-Hilbert problem. We demonstrate the validity and applicability of the method by computing the Anderson orthogonality catastrophe in the $c\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, corresponding to free fermions.