Lower Spectral Branches of a Particle Coupled to a Bose Field

TL;DR

This paper analyzes the lower spectral branches of the Fröhlich polaron Hamiltonian in dimensions d≥3, revealing a single polaron branch and a manifold of polaron+boson states, with detailed dispersion laws and eigenfunctions.

Contribution

It provides a detailed spectral analysis of the polaron Hamiltonian's lower spectrum, including dispersion laws and eigenfunctions, in the weak coupling regime for dimensions d≥3.

Findings

Single polaron branch exists within a bounded momentum domain.

Polaron dissolves into polaron+boson states at the domain boundary.

Explicit dispersion laws and eigenfunctions are derived.

Abstract

The structure of the lower part (i.e. $\epsilon $-away below the two-boson threshold) spectrum of Fr\"ohlich's polaron Hamiltonian in the weak coupling regime is obtained in spatial dimension $d\geq 3$. It contains a single polaron branch defined for total momentum $p\in G^{(0)} $, where $G^{(0)}\subset {\mathbb R}^d$ is a bounded domain, and, for any $p\in {\mathbb R}^d$, a manifold of polaron + one-boson states with boson momentum $q$ in a bounded domain depending on $p$. The polaron becomes unstable and dissolves into the one boson manifold at the boundary of $G^{(0)}$. The dispersion laws and generalized eigenfunctions are calculated.