Geometric methods for nonlinear many-body quantum systems

TL;DR

This paper introduces geometric methods for analyzing nonlinear many-body quantum systems, extending classical spectral techniques to nonlinear regimes and providing new results on Hartree-Fock states and multi-polaron existence.

Contribution

It develops a formalism using a weak topology on many-body states to study nonlinear systems, and applies it to prove nonlinear HVZ theorems and existence of multi-polaron states.

Findings

Proved a nonlinear version of the HVZ theorem.

Analyzed geometric properties of Hartree-Fock states.

Established existence of multi-polaron in the Pekar-Tomasevich model.

Abstract

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schr\"odinger operators. In this paper we provide a formalism which also allows to study nonlinear systems. We start by defining a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. We provide several important properties of this topology and use them to provide a simple proof of the famous HVZ theorem in the repulsive case. In a second step we recall the method of geometric localization in Fock space as proposed by Derezi\'nski and G\'erard, and we relate this tool to our weak topology. We then provide several applications. We start by studying the so-called finite-rank approximation which consists in imposing that the many-body wavefunction can be expanded using finitely many one-body functions. We thereby emphasize geometric properties of Hartree-Fock states and prove nonlinear versions of the HVZ theorem, in the spirit of works of Friesecke. In the last section we study translation-invariant many-body systems comprising a nonlinear term, which effectively describes the interactions with a second system. As an example, we prove the existence of the multi-polaron in the Pekar-Tomasevich approximation, for certain values of the coupling constant.