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We derive an effective equation for the dynamics of many identical bosons indimension one, in the presence of a tiny impurity. The interaction betweenevery pair of bosons is mediated by the impurity through a three-bodyinteraction. Assuming a simultaneous mean-field and short-range scaling, withthe short-range proceeding slower than the mean-field, and choosing an initialfully condensed state, we prove propagation of chaos and obtain an effectiveone-particle Schr\"odinger equation with a nonlinearity concentrated at apoint. More precisely, we prove convergence of one-particle density operatorsin the trace-class topology, and estimate the fluctuations as superexponential.This is the first derivation of the so-called nonlinear delta model, widelyinvestigated in the last decades.

Microscopic derivation of a Schr\"odinger equation in dimension one with a nonlinear point interaction