On the derivation of the Schrödinger equation with point-like nonlinearity
https://doi.org/10.17586/2220-8054-2015-6-1-79-94
Abstract
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 Schrodinger equation,
nonlinear delta interactions,
zero-range limit of concentrated nonlinearities
Supporting agencies: The author is grateful to his friends and collaborators Domenico Finco, Diego Noja, and Alessandro Teta, all the results and conjectures presented in this report were obtained and formulated in collaboration with them. The support of the FIR 2013 project \Condensed Matter in Mathematical Physics" (code RBFR13WAET) is also acknowledged.
About the Author
C. Cacciapuoti
Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria
Italy
Italy
Via Valleggio 11, 22100 Como
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Review
For citations:
Cacciapuoti C. On the derivation of the Schrödinger equation with point-like nonlinearity. Nanosystems: Physics, Chemistry, Mathematics. 2015;6(1):79-94. https://doi.org/10.17586/2220-8054-2015-6-1-79-94
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