Geometric methods for nonlinear many-body quantum systems

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.Comment: Final version to appear in Journal of Functional Analysi

Symmetry of Bipolaron Bound States for Small Coulomb Repulsion

“…Moreover, simple trial function arguments show that e U ≤ 2e and e symm U ≤ e for all U . Lewin [12] has shown that the infimum e U is attained provided e U < 2e. In the Appendix of this paper we shall prove an analogous result for the rotation invariant problem, with a different condition, however, namely, e symm U < e.…”

“…Since the minimizer for this problem is unique up to translations (see Step 2 above), the results of [12] imply that…”

“…A minimizer also exists for a bipolaron provided the energy is below the energy of twice the single polaron energy [12]. Interestingly, the bipolaron has a minimizer with finite radius at the critical value U = U c [4].…”

Derivation of Hartreeʼs theory for generic mean-field Bose systems

“…The weak version of the quantum de Finetti theorem is related to recent results of Ammari and Nier [3,4] (see Section 2 for a more precise discussion), but we follow a different approach, based on geometric methods for many-body systems [36]. Our proof of the weak quantum de Finetti theorem is in fact only the first step towards a more precise understanding of the lack of compactness for general systems and we will repeatedly use more refined arguments in the paper.…”

“…The set P (k) w somehow only describe the particles which have not escaped, and the information on the other ones is completely lost. The accurate description of the lack of compactness will be done in this article, using the geometric methods of [36]. These couple the (somehow algebraic) properties of many-particle systems with techniques from nonlinear analysis in the spirit of the concentration-compactness theory [39,47].…”