A strong operator topology adiabatic theorem

TL;DR

This paper establishes an adiabatic theorem for spectral data evolution under weak perturbations in systems lacking an intrinsic time scale, with convergence types depending on the function class.

Contribution

It introduces a new adiabatic theorem applicable to systems without intrinsic time scales, covering a broader class of spectral functions and projections.

Findings

Norm convergence for continuous functions of the Hamiltonian.

Strong operator topology convergence for spectral projections, including embedded eigenvalues.

Applicable to systems with weak additive perturbations.

Abstract

We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topology.