Spectral Analysis of a Model for Quantum Friction

TL;DR

This paper investigates the spectral properties of a quantum Hamiltonian model for quantum friction, showing the absence of a ground state at non-zero momentum and the presence of absolutely continuous spectrum.

Contribution

It provides a spectral analysis of a quantum model for friction, extending classical models and revealing the spectral structure under certain infrared conditions.

Findings

No ground state exists at non-zero total momentum.

Spectrum is absolutely continuous except at zero momentum.

Quantum and classical friction models are compared.

Abstract

An otherwise free classical particle moving through an extended spatially homogeneous medium with which it may exchange energy and momentum will undergo a frictional drag force in the direction opposite to its velocity with a magnitude which is typically proportional to a power of its speed. We study here the quantum equivalent of a classical Hamiltonian model for this friction phenomenon that was proposed in [11]. More precisely, we study the spectral properties of the quantum Hamiltonian and compare the quantum and classical situations. Under suitable conditions on the infrared behaviour of the model, we prove that the Hamiltonian at fixed total momentum has no ground state except when the total momentum vanishes, and that its spectrum is otherwise absolutely continuous.