Finite-well potential in the 3D nonlinear Schroedinger equation: Application to Bose-Einstein condensation

TL;DR

This paper investigates the existence and stability of bound states in the 3D nonlinear Schrödinger equation with a finite square-well potential, relevant for Bose-Einstein condensates, revealing conditions for stability, collapse, and unbound states.

Contribution

It demonstrates the existence of normalizable negative-energy bound states in 3D nonlinear Schrödinger equations with finite wells, including stability criteria and potential experimental realization.

Findings

Normalizable negative-energy bound states exist for certain nonlinearity ranges.

System becomes unstable and collapses below a critical attractive nonlinearity.

Highly repulsive nonlinearity prevents bound state formation.

Abstract

Using variational and numerical solutions we show that stationary negative-energy localized (normalizable) bound states can appear in the three-dimensional nonlinear Schr\"odinger equation with a finite square-well potential for a range of nonlinearity parameters. Below a critical attractive nonlinearity, the system becomes unstable and experiences collapse. Above a limiting repulsive nonlinearity, the system becomes highly repulsive and cannot be bound. The system also allows nonnormalizable states of infinite norm at positive energies in the continuum. The normalizable negative-energy bound states could be created in BECs and studied in the laboratory with present knowhow.