The nonlinear Schrödinger equation on a lattice — a quantum model of interacting particles hopping between discrete sites — has a ground state described by an integral equation. The integral equation encodes the distribution of quantum numbers that characterize the ground state. For finite coupling strength, this distribution is smooth, bounded, and unremarkable. It is a technical object solved by standard methods.
Prolhac (arXiv:2603.09522, March 2026) pushes the coupling to zero and finds that the integral equation becomes doubly singular — both the driving term and the integral kernel collapse to delta functions. The equation doesn't simplify. It breaks, in a way that requires matched asymptotic analysis across three regions (inner, outer, and edge) to resolve. The inner region, rescaled to absorb the singularity, yields a solution whose Fourier transform is exactly the Bose-Einstein distribution.
This is not an analogy. The lattice nonlinear Schrödinger model at zero coupling reduces to free bosons, and the ground-state quantum number distribution, properly rescaled, becomes the occupation number distribution of an ideal Bose gas. The Bose-Einstein distribution — usually derived from maximizing entropy subject to particle number and energy constraints — emerges here from the singular limit of a Bethe ansatz integral equation. The statistical mechanics was hiding inside the integrable structure.
The edge region, where the inner and outer solutions must match, introduces exponentially small corrections described by Wiener-Hopf factorization. These corrections form a resurgent transseries — a mathematical structure where the perturbative expansion (power series in the coupling) and nonperturbative contributions (exponentially small terms) are locked together by precise relations. The instanton action governing these corrections connects to the Love integral equation, which describes the capacitance of a circular disc. A problem in quantum many-body physics, in its singular limit, borrows the mathematics of electrostatics.
The general observation: smooth equations at finite parameters can contain, compressed into their singular limits, the complete structure of an apparently unrelated physical theory. The Bose-Einstein distribution wasn't added to the lattice NLS equation. It was already there, invisible at any finite coupling, visible only when the coupling was pushed to the point where the equation itself nearly ceased to exist. The singularity doesn't destroy information — it reveals information that was encoded in the equation's structure all along, inaccessible to any finite-coupling analysis.
Push a wave equation to its breaking point, and you find a census of particles waiting in the wreckage.
Prolhac, "Weak-Coupling Limit of the Lattice Nonlinear Schrödinger Integral Equation," arXiv:2603.09522 (March 2026).