On the Dynamics of Crystal Electrons, high Momentum Regime

TL;DR

This paper investigates the quantum behavior of electrons in crystal lattices under a specific Hamiltonian, proving the continuous spectrum is always present and that small potentials do not produce bound states.

Contribution

It establishes the non-emptiness of the continuous spectrum and absence of bound states for small periodic potentials in the studied quantum model.

Findings

Continuous spectrum of the Hamiltonian is never empty.

No bound states exist when the potential is sufficiently weak.

Results apply to quantum dynamics in crystal electron models.

Abstract

We study the quantum dynamics generated by $H^{SW}=-{d^2\over dx^2}+V-x$ with $V$ a real periodic function of weak regularity. We prove that the continuous spectrum of $H^{SW}$ is never empty, and furthermore that for $V$ small enough there are no bound states.