A microscopic derivation of time-dependent correlation functions of the $1D$ cubic nonlinear Schr\"{o}dinger equation

TL;DR

This paper derives time-dependent correlation functions of the 1D cubic nonlinear Schr"odinger equation from many-body quantum theory, addressing challenges posed by particle number cutoffs and extending results to nonlocal and local NLS cases.

Contribution

It provides a microscopic derivation of correlation functions for 1D cubic NLS, overcoming technical obstacles and applying complex analytic methods to both nonlocal and local cases.

Findings

Derived time-dependent correlation functions from many-body quantum theory.

Extended methods to nonlocal NLS with bounded convolution potential.

Analyzed local NLS using periodic Strichartz estimates in $X^{s,b}$ spaces.

Abstract

We give a microscopic derivation of time-dependent correlation functions of the $1D$ cubic nonlinear Schr\"{o}dinger equation (NLS) from many-body quantum theory. The starting point of our proof is our previous work on the time-independent problem and work of the second author on the corresponding problem on a finite lattice. An important new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaussian structure of the free quantum field and prevents the use of the Wick theorem. We overcome it by the means of complex analytic methods. Our methods apply to the nonlocal NLS with bounded convolution potential. In the periodic setting, we also consider the local NLS, arising from short-range interactions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in $X^{s,b}$ spaces.