A 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 write 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 $c\to \mathrm{\infty }$ limit, corresponding to free fermions.