Weakly nonlinear time-adiabatic theory

TL;DR

This paper extends the quantum time-adiabatic theorem to weakly nonlinear Schrödinger equations, demonstrating stability of linear and nonlinear bound states under small nonlinear effects.

Contribution

It introduces a notion of criticality for weakly nonlinear regimes and proves adiabatic stability of bound states in this context.

Findings

Linear bound states remain adiabatically stable.

Nonlinear effects manifest as a slowly varying phase modulation.

Certain nonlinear bound states also stay adiabatically stable.

Abstract

We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schroedinger equations in which either the nonlinear coupling constant or, equivalently, the solution is asymptotically small. To this end, a notion of criticality is introduced at which the linear bound states stay adiabatically stable, but nonlinear effects start to show up at leading order in the form of a slowly varying nonlinear phase modulation. In addition, we prove that in the same regime a class of nonlinear bound states also stays adiabatically stable, at least in terms of spectral projections.