Weak-Coupling Limit of the Lattice Nonlinear Schr\"odinger Integral Equation

We study the ground-state integral equation of the quantum lattice nonlinear Schr\"odinger model -- equivalently the isotropic Heisenberg XXX spin chain with spin $s = -1$ -- in the weak-coupling limit. Unlike the continuous Lieb--Liniger equation, whose driving term is a constant, the lattice equation is doubly singular: both the driving term and the integral kernel degenerate into $\delta$-functions as $\kappa \to 0$. We develop a matched asymptotic expansion with three regions -- inner, outer, and edge. We show that the Fourier transform of the rescaled inner solution is exactly the Bose--Einstein distribution, and the peak density diverges logarithmically with a constant $C$, which we determine analytically via two independent routes and confirm numerically. A duality with the Love integral equation for the circular disc capacitor yields the total density expansion. We prove an identity for the inner energy, allowing us to obtain the ground-state energy per site. From the Wiener--Hopf factorisation of the edge boundary layer, we identify the instanton action and predict a resurgent transseries structure.