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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; claudio.cacciapuoti@uninsubria.it

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 [10], 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

DOI 10.17586/2220-8054-2015-6-1-79-94

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