# FWF Grant

# Project Details

Type: FWF Individual Grant

Duration: April 1, 2015 - March 31, 2018

Granted money: EUR 315K

Project Nr. P 27533-N27

# Abstract

The main focus of this research 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 can thus increase the understanding of physical systems.

From the point of view of mathematical physics, there has been substantial progress in the last few years in understanding some of the interesting phenomena occurring in quantum gases, and the goal of this project is to further investigate some of the relevant issues. Due to the complex nature of the problems, new mathematical ideas and methods will have to be developed for this purpose. Progress along these lines can be expected to yield valuable insight into the complex behavior of many-body quantum systems at low temperature.

The main question addressed in this research 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 ranges over the whole system size. Among the questions that are addressed in this project are the extension of these results to the physically more relevant case of short-range interactions, to the case of rotating systems, and to the study of the structure of the excitation spectrum in the thermodynamic limit.

# Publications

- R.L Frank, R. Killip, P.T. Nam, Nonexistence of large nuclei in the liquid drop model, Lett. Math. Phys. 106, 1033 (2016).
- R.L. Frank, P.T. Nam, H. Van Den Bosch, The maximal excess charge in Müller density-matrix-functional theory, preprint arXiv:1608.05625
- R.L. Frank, P.T. Nam, H. Van Den Bosch, The ionization conjecture in Thomas-Fermi-Dirac-von Weizsäcker theory, preprint arXiv:1606.07355
- X. Li, R. Seiringer, M. Lemeshko, Angular self-localization of impurities rotating in a bosonic bath, preprint arXiv:1610.04908
- T. Moser, R. Seiringer, Stability of a fermionic N + 1 particle system with point interactions, preprint arXiv:1609.08342
- T. Moser, R. Seiringer, Triviality of a model of particles with point interactions in the thermodynamic limit, Lett. Math. Phys. 107, 533 (2017)
- P.T. Nam, M. Napiorkowski, A note on the validity of Bogoliubov correction to mean-field dynamics, preprint arXiv:1604.05240
- P.T. Nam, H. Van Den Bosch, Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker theory with small nuclear charges, Math. Phys. Anal. Geom. 20:6 (2017)
- M. Correggi, A. Giuliani, R. Seiringer, Low-Temperature Spin-Wave Approximation for the Heisenberg Ferromagnet, preprint, arXiv:1602.00155
- M. Napiorkowski, R. Reuvers, J.P. Solovej, The Bogoliubov free energy functional I. Existence of minimizers and phase diagram, preprint arXiv:1511.05935
- M. Napiorkowski, R. Reuvers, J.P. Solovej, The Bogoliubov free energy functional II. The dilute limit, preprint arXiv:1511.05953
- R. Seiringer, S. Warzel, Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas, New J. Phys. 18, 035002 (2016).
- P.T. Nam, M. Napiorkowski, Bogoliubov correction to the mean-field dynamics of interacting bosons, preprint arXiv:1509.04631
- M. Koenenberg, T.Moser, R. Seiringer, J. Yngvason, Superfluidity and BEC in a Model of Interacting Bosons in a Random Potential, J. Phys.: Conf. Ser. 691, 012016 (2016).
- P.T. Nam, M. Napiorkowski, J.P. Solovej, Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations, J. Funct. Anal. 270, 4340 (2016).
- G. Braeunlich, C. Hainzl, R. Seiringer, Bogolubov–Hartree–Fock theory for strongly interacting fermions in the low density limit, Math. Phys. Anal. Geom. 19:13 (2016).
- A. Giuliani, R. Seiringer, Periodic striped ground states in Ising models with competing interactions, Commun. Math. Phys. 347, 983 (2016)
- R.L. Frank, C. Hainzl, B. Schlein, R. Seiringer, Incompatibility of time-dependent Bogoliubov–de-Gennes and Ginzburg–Landau equations, Lett. Math. Phys. 106, 913 (2016).

# Former project members

Marcin Napiorkowski (PostDoc)

Phan Thanh Nam (PostDoc)