On uniqueness and non-degeneracy of anisotropic polarons

TL;DR

This paper investigates the anisotropic Choquard--Pekar equation, establishing the uniqueness, non-degeneracy, and symmetry properties of minimizers in anisotropic media, with implications for understanding polaron behavior.

Contribution

It proves the uniqueness and non-degeneracy of minimizers and characterizes their symmetry properties in anisotropic media, extending previous isotropic results.

Findings

Uniqueness of minimizers in weakly anisotropic media

Non-degeneracy of these minimizers

Symmetry properties of minimizers and kernel structure

Abstract

We study the anisotropic Choquard--Pekar equation which de-scribes a polaron in an anisotropic medium. We prove the uniqueness and non-degeneracy of minimizers in a weakly anisotropic medium. In addition, for a wide range of anisotropic media, we derive the symmetry properties of minimizers and prove that the kernel of the associated linearized operator is reduced, apart from three functions coming from the translation invariance, to the kernel on the subspace of functions that are even in each of the three principal directions of the medium.