Geometric methods for nonlinear many-body quantum systems. (English) Zbl 1216.81180
The author presents a geometric formalism able to handle particular nonlinear problems appearing in quantum mechanics of many particle systems, specially when some of these particles are allowed to escape at infinity. For that purpose a suitable weak topology is introduced leading to the notion of geometric convergence involving state spaces with variable dimensions. As an application, an elegant proof of the HVZ (Hunziker-Van Winter-Zhislin) theorem is provided. Links of this geometric convergence notion with the geometrical localization method introduced by Dereziński and Gerard are then clarified. Several applications are finally provided: the finite-rank approximation (known as Multi Configuration Approximation in quantum chemistry), the analysis of some translation-invariant systems involving non-linear effective interactions (appearing in low energy Nuclear Physics) and the multi-polaron system, an effective model in electron-phonon interaction.
Reviewer: Bernard Ducomet (Bruyères le Châtel)
MSC:
| 81V70 | Many-body theory; quantum Hall effect |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
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