NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2015, 6 (1), P. 79-94
ON THE DERIVATION OF THE SCHRÖDINGER EQUATION WITH POINT-LIKE NONLINEARITY
C. Cacciapuoti – Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy; email@example.com
In this report we discuss the problem of approximating nonlinear delta-interactions in dimensions one and three with regular, local or non-local nonlinearities. Concerning the one dimensional case, we discuss a recent result proved in , on the derivation of nonlinear delta-interactions as limit of scaled, local nonlinearities. For the three dimensional case, we consider an equation with scaled, non-local nonlinearity. We conjecture that such an equation approximates the nonlinear delta-interaction, and give an heuristic argument to support our conjecture.
Keywords: Nonlinear Schrödinger equation, nonlinear delta interactions, zero-range limit of concentrated nonlinearities.
PACS 02.30.Jr, 03.65.Db, 02.30.Rz