Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction
Abstract
We derive an effective equation for the dynamics of many identical bosons in dimension one in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a positive threebody potential. Assuming a simultaneous meanfield and shortrange scaling with the shortrange proceeding slower than the meanfield, and choosing an initial fully condensed state, we prove propagation of chaos and obtain an effective oneparticle Schrödinger equation with a defocusing nonlinearity concentrated at a point. More precisely, we prove convergence of oneparticle density operators in the traceclass topology and estimate the fluctuations as superexponential. This is the first derivation of the socalled nonlinear delta model, widely investigated in the last decades.
 Publication:

arXiv eprints
 Pub Date:
 August 2023
 DOI:
 10.48550/arXiv.2308.09674
 arXiv:
 arXiv:2308.09674
 Bibcode:
 2023arXiv230809674A
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs