# ERC Grant

# Project Details

Type: ERC Advanced Grant

Duration: October 1, 2016 - September 30, 2021

Granted money: EUR 1498K

Project Nr. 694227 (AQUAMS)

# Abstract

The main focus of this project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g., Bose--Einstein condensation and superfluidity, which makes them interesting both from a physical and a mathematical point of view.

The goal of this project is the development of new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and thus increase our understanding of physical systems. From the point of view of mathematical physics, there has been significant progress in the last few years in understanding the interesting phenomena occurring in quantum gases, and the goal of this project is to investigate some of the key issues that remain unsolved. Due to the complex nature of the problems, new mathematical ideas and methods will have to be developed for this purpose.

One of the main question addressed in this proposal is the validity of the Bogoliubov approximation for the excitation spectrum of many-body quantum systems. While its accuracy has been successfully shown for the ground state energy of various models, its predictions concerning the excitation spectrum have so far only been verified in the Hartree limit, an extreme form of a mean-field limit where the interaction among the particles is very weak and ranges over the whole system. The central part of this project is concerned with the extension of these results to the case of short-range interactions. Apart from being mathematically much more challenging, the short-range case is the one most relevant for the description of actual physical systems. Hence progress along these lines can be expected to yield valuable insight into the complex behavior of these many-body quantum systems.

# Publications

- S. Mayer, R. Seiringer, The free energy of the two-dimensional dilute Bose gas. II. Upper bound, preprint, arXiv:2002.08281
- R. Seiringer, J. Yngvason, Emergence of Haldane pseudo-potentials in systems with short-range interactions, preprint, arXiv:2001.07144
- N. Leopold, D. Mitrouskas, R. Seiringer, Derivation of the Landau–Pekar equations in a many-body mean-field limit, preprint, arXiv:2001.03993
- R. Seiringer, The Polaron at Strong Coupling, preprint, arXiv:1912.12509
- E. Yakaboylu, A. Ghazaryan, D. Lundholm, N. Rougerie, M. Lemeshko, R. Seiringer, A Quantum Impurity Model for Anyons, preprint, arXiv:1912.07890
- M. Napiorkowski, R. Seiringer, Free energy asymptotics of the quantum Heisenberg spin chain, preprint, arXiv:1912.01967
- A. Deuchert, S. Mayer, R. Seiringer, The free energy of the two-dimensional dilute Bose gas. I. Lower bound, preprint, arXiv:1910.03372
- N. Leopold, S. Rademacher, B. Schlein, R. Seiringer, The Landau–Pekar equations: Adiabatic theorem and accuracy, preprint, arXiv:1904.12532
- D. Feliciangeli, R. Seiringer, Uniqueness and Non-degeneracy of Minimizers of the Pekar Functional on a Ball, SIAM J. Math. Anal. 52, 605 (2020).
- M. Lewin, E.H. Lieb, R. Seiringer, Floating Wigner crystal with no boundary charge fluctuations, Phys. Rev. B 100, 035127 (2019)
- A.Deuchert, R. Seiringer, Gross-Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature, preprint, arXiv:1901.11363, Arch. Rat. Mech. Anal. (in press)
- R.L. Frank, R. Seiringer, Quantum corrections to the Pekar asymptotics of a strongly coupled polaron, preprint, arXiv:1902.02489, Commun. Pure Appl. Math. (in press)
- E.H. Lieb, R. Seiringer, Divergence of the effective mass of a polaron in the strong coupling limit, preprint, arXiv:1902.04025, J. Stat. Phys. (in press)
- M. Lewin, E.H. Lieb, R. Seiringer, The local density approximation in density functional theory, Pure Appl. Anal. 2, 35 (2020)
- N. Benedikter, P.T. Nam, M. Porta, B. Schlein, R. Seiringer, Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime, preprint, arXiv:1809.01902, Commun. Math. Phys. (in press)
- A. Deuchert, R. Seiringer, J. Yngvason, Bose-Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature, Commun. Math. Phys. 368, 723 (2019)
- E. Yakaboylu, B. Midya, A. Deuchert, N. Leopold, M. Lemeshko, Theory of the rotating polaron: Spectrum and self-localization, Phys. Rev. B 98, 224506 (2018)
- T. Moser, R. Seiringer, Energy contribution of a point interacting impurity in a Fermi gas, Ann. Henri Poincaré 20, 1325 (2019)
- T. Moser, R. Seiringer, Stability of the 2+2 fermionic system with point interactions, Math. Phys. Anal. Geom. 21, Art. 19 (2018)
- D. Lundholm, R. Seiringer, Fermionic behavior of ideal anyons, preprint, Lett. Math. Phys. 108, 2523–2541 (2018)
- M. Lewin, E.H. Lieb, R. Seiringer, Statistical Mechanics of the Uniform Electron Gas, Journal de l’Ecole polytechnique - Mathematiques, Tome 5, 79 (2018)
- A. Deuchert, A. Geisinger, C. Hainzl, M. Loss, Persistence of translational symmetry in the BCS model with radial pair interaction, Ann. H. Poincaré 19, 1507 (2018)
- E. Yakaboylu, A. Deuchert, M. Lemeshko, Emergence of non-abelian magnetic monopoles in a quantum impurity problem, Phys. Rev. Lett. 119, 235301 (2017)
- A. Deuchert, A lower bound for the BCS functional with boundary conditions at infinity, J. Math. Phys. 58, 081901 (2017)
- X. Li, R. Seiringer, M. Lemeshko, Angular self-localization of impurities rotating in a bosonic bath, Phys. Rev. A 95, 033608 (2017)
- T. Moser, R. Seiringer, Stability of a fermionic N + 1 particle system with point interactions, Commun. Math. Phys. 356, 329 (2017)

# Former project members

Niels Benedikter (PostDoc)

Andreas Deuchert (PostDoc)

Nikolai Leopold (PostDoc)

Simon Mayer (PhD Student)

Dario Feliciangeli (PhD Student)