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.)(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.