Nonlinear time-dependent one-dimensional Schroedinger equation with double well potential

TL;DR

This paper analyzes the nonlinear time-dependent one-dimensional Schrödinger equation with a double well potential, demonstrating how nonlinearity affects quantum beating motion and providing precise error estimates for a 2-mode approximation.

Contribution

It introduces a rigorous reduction of the nonlinear Schrödinger equation to a 2-mode model in the semiclassical limit, with stability analysis and error bounds.

Findings

2-mode approximation accurately describes the solution

Nonlinearity destroys the quantum beating motion

Error estimates validate the approximation

Abstract

We consider time-dependent Schroedinger equations in one dimension with double well potential and an external nonlinear perturbation. If the initial state belongs to the eigenspace spanned by the eigenvectors associated to the two lowest eigenvalues then, in the semiclassical limit, we show that the reduction of the time-dependent equation to a 2-mode equation gives the dominant term of the solution with a precise estimate of the error. By means of this stability result we are able to prove the destruction of the beating motion for large enough nonlinearity.