Adiabatic Evolution for Systems with Infinitely many Eigenvalue Crossings

TL;DR

This paper develops an adiabatic theorem for quantum systems with infinitely many eigenvalue crossings, providing bounds on correction terms and requiring minimal regularity assumptions.

Contribution

It introduces a new adiabatic theorem applicable to models with infinitely many eigenvalue crossings, expanding the theoretical framework for such systems.

Findings

Provides an upper bound on correction terms in the adiabatic limit.

Requires only differentiability of spectral projectors and geometric conditions at crossings.

Extends adiabatic theory to complex spectral crossing scenarios.

Abstract

We formulate an adiabatic theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. We give an upper bound on the leading correction terms with respect to the adiabatic limit. The result requires only differentiability of the considered spectral projector, and some geometric hypothesis on the local behaviour of the eigenvalues at the crossings.