Adiabatic Theorem in the Case of Continuous Spectra

TL;DR

This paper rigorously proves the quantum adiabatic theorem for systems with continuous spectra by introducing a new concept of a virtual gap and establishing the conditions for adiabatic approximation validity.

Contribution

It introduces a novel approach using eigendifferentials and differential projectors to extend the adiabatic theorem to continuous spectra systems.

Findings

Defined a 'virtual gap' for continuous spectra

Derived the validity condition for adiabatic approximation

Provided a rigorous proof for non-degenerate continuous spectra

Abstract

In this paper,we present a rigorous demonstration and discussion of the quantum adiabatic theorem for systems having a non degenerate continuous spectrum. A new strategy is initiated by defining a kind of gap, "a virtual gap", for the continuous spectrum through the notion of eigendifferential (Weyl's packet) and using the differential projector operator. Finally we obtain the validity condition of the adiabatic approximation.