# A Nonlinear Quantum Adiabatic Approximation

## TL;DR

This paper extends the quantum adiabatic theorem to nonlinear Hamiltonians, proving the existence of solutions that closely follow instantaneous nonlinear eigenvectors during slow Hamiltonian variations.

## Contribution

It introduces a nonlinear generalization of the quantum adiabatic theorem, establishing conditions for solutions to remain near nonlinear eigenstates.

## Key findings

- Existence of solutions close to nonlinear eigenvectors in adiabatic regime
- Extension of results from bounded to unbounded Hamiltonians under spectral assumptions
- Introduction of a framework for nonlinear quantum adiabatic approximation

## Abstract

This paper is devoted to a generalisation of the quantum adiabatic theorem to a nonlinear setting. We consider a Hamiltonian operator which depends on the time variable and on a finite number of parameters and acts on a separable Hilbert space of which we select a fixed basis. We study an evolution equation in which this Hamiltonian acts on the unknown vector, while depending on coordinates of the unknown vector in the selected basis, thus making the equation nonlinear. We prove existence of solutions to this equation and consider their asymptotics in the adiabatic regime, i.e. when the Hamiltonian is slowly varying in time. Under natural spectral hypotheses, we prove the existence of normalised time dependent vectors depending on the Hamiltonian only, called instantaneous nonlinear eigenvectors, and show the existence of solutions which remain close to these vectors, up to a rapidly oscillating phase, in the adiabatic regime. We first investigate the case of bounded operators and then exhibit a set of spectral assumptions under which the result extends to unbounded Hamiltonians.