Adiabatic Theorem without a Gap Condition

TL;DR

This paper proves an adiabatic theorem for quantum systems without requiring a spectral gap, broadening its applicability to systems like atoms in quantized radiation fields, but does not specify the convergence rate.

Contribution

It establishes the adiabatic theorem under weaker conditions, needing only a piecewise twice differentiable spectral projection, unlike traditional gap-dependent theorems.

Findings

Validates the adiabatic theorem without a spectral gap.

Applicable to ground states of atoms in quantized radiation fields.

Provides a foundation for estimating convergence rates with additional spectral info.

Abstract

We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.