The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension:(Formula presented.)This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit (Formula presented.) the weak (integral) dynamics converges in (Formula presented.) to the weak dynamics of the NLS with point-concentrated nonlinearity:(Formula presented.)where Hα is the Laplacian with the nonlinear boundary condition at the origin (Formula presented.) and (Formula presented.). The convergence occurs for every (Formula presented.) if V ≥ 0 and for every (Formula presented.) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points