Summer School
June 19+22, 2020

# Applications of Bogoliubov Theory

## June 19 + 22, 2020 (Friday and Monday)

### Concept

The cancellation of most events this summer means that PhD students lose opportunities to learn about topics outside their home working group and to get to know other students and researchers. The main lectures in this summer school are an opportunity to learn about the mathematics of many-body quantum systems. Furthermore, all participants are invited to present their research in small online groups to get to know each other and chat informally ("virtual post session").

For whom? Students and researchers in mathematical physics from all directions (not necessarily quantum systems).

Lectures by: Marcin Napiórkowski (University of Warsaw), Benjamin Schlein (University of Zurich), Robert Seiringer (IST Austria), Jan Philip Solovej (University of Copenhagen)

Virtual poster session: open to all interested participants (not necessarily working on quantum systems). Younger participants are particularly encouraged to participate. Please read carefully the poster session concept before preparing your poster.

### Program

You may download the slides of the lectures here: Solovej, Seiringer, Napiórkowski, Schlein.

On both days, the program starts in the early afternoon (Italian time). The first afternoon concludes with a virtual poster session.

Friday, June 19, 2020: 14:00-15:00 lecture Solovej, 15:15-16:15 lecture Seiringer, 16:30-18:00 virtual poster session

Monday, June 22, 2020: 14:00-15:00 lecture Napiórkowski, 15:15-16:15 lecture Schlein

Please find below the abstracts of the lectures:

Jan Philip Solovej: Many body Bose gases and the Bogolubov approximation
I will give the basics of many-body quantum mechanics and in particular the case for Bose gases. I will introduce the method of second quantization. I will then discuss Bogolubov's approximation in his original formulation, where the many-body Hamiltonian is approximated by what is called a quadratic Hamiltonian. Quadratic Hamiltonians have equilibrium states that are quasi-free or Gaussian states characterized by satisfying Wick's relation. I will use this to reformulate the Bogolubov approximation in terms of a variational model.

Robert Seiringer: The excitation spectrum of weakly interacting bosons
Of particular relevance for an understanding of the low-temperature properties of a quantum system is the structure of the excitation spectrum, which is the part of the spectrum close to the spectral bottom. We present recent progress on this question for bosonic many-body quantum systems with weak two-body interactions. We investigate the accuracy of the Bogoliubov approximations, which predicts that the low-energy spectrum is made up of sums of elementary excitations, with linear dispersion law at low momentum. The latter property is crucial for the superfluid behavior the system.

Marcin Napiórkowski: Bogoliubov Theory at Positive Temperatures
I shall discuss the homogeneous Bose gas at positive temperatures within Bogoliubov theory. The theory arises by restricting the Hilbert space to quasi-free states. I will introduce the free energy functional and discuss the existence of equilibrium states, phase diagram and critical temperature. Based on joint work with Robin Reuvers and Jan Philip Solovej.

Benjamin Schlein: Dynamics of a strongly coupled polaron
We are interested in the time-evolution of a Fröhlich polaron, describing an electron coupled to a quantised phonon field, with a large coupling constant $\alpha$. For product initial data, with a coherent cloud of phonons and with the electron minimising the corresponding energy, we construct an approximation of the evolution, valid up to times of order $\alpha^2$. The approximation is given in terms of a product state, evolved through the classical Landau-Pekar equations, adjusted by a Bogoliubov dynamics, taking quantum fluctuations into account. This is joint work with N. Leopold, D. Mitrouskas, S. Rademacher and R. Seiringer.

### Registration

Registration with an email address is necessary to control access to the zoom channels. Please fill in the registration form (registration is closed now).

### Zoom

The entire program will be held using Zoom. If you participate in the poster session or give a lecture, you will need to create your own (free) Zoom account, install or update the app, and accomodate yourself with its operation before the program.

Lectures: All main lectures are held in the zoom meeting ID: 973 3987 5546. The password will be provided by email to registered participants.

Virtual poster session: All participants who present a poster need to open their own Zoom channel for the time of their poster session (free zome basic account needed). To avoid interference by trolls, it is important that you set the password as communicated by email (same password as for the main lectures).

• Konstantin Merz: Weakly coupled bound states of $|p^2-1|-\lambda V$. ID: 635 806 7115
• Jinyeop Lee: Rate of convergence towards equations of Hartree type for mixture condensates with factorized initial data. ID: 594 429 4003
• Théotime Girardot: Average Field Approximation for Almost Fermionic Anyons. ID: 992 1831 0933
• Krzysztof Myśliwy: Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron. ID: 409 378 4369
• Alessandro Olgiati: Localization regime for bosons in a double well. ID: 733 2033 2330

If you are presenting a poster using a free zoom basic account, your channel will automatically close after 40 minutes - just reopen it, participants will be able to reconnect with the same link/ID.

### Participants

Please find below the list of registered participants who have agreed to their name being published.

1. Chiara Boccato, IST Austria
2. Krzysztof Myśliwy, IST Austria
3. Arnaud Triay, LMU - München
4. Oluwadara Ogunkoya, University of Alabama at Birmingham
5. Daniele Dimonte, University of Basel
6. Serena Cenatiempo, Gran Sasso Science Institute
7. Jacob Fronk, Heidelberg University/University of Copenhagen
8. Dario Feliciangeli, IST Austria
9. Francesco Raso Stoia, Universitá degli Studi di Milano Bicocca
10. Lea Boßmann, IST Austria
11. Jinyeop Lee, Korea Institute for Advanced Study
12. Gianfausto Dell'Antonio, Sissa
13. Elliott Lieb, Princeton
14. Martin Ravn Christiansen, LMU Munich
15. Gustavo de Oliveira, Universidade Federal de Minas Gerais
16. Gianluca Panati, "La Sapienza" University of Rome
17. Ludwik Dabrowski, SISSA, Trieste
18. Phan Thanh Nam, LMU Munich
19. Paolo Facchi, Universita di Bari, Italy
20. Daniele Valeri, University of Glasgow
21. Giuseppe Gaeta, Università degli Studi di Milano
22. Giuseppe Bevilacqua, Università di Siena
23. Antonio Moscato, Trento University
24. Chun Yin Lam, LMU Muenchen
25. Nicolas Rougerie, Université Grenoble-Alpes & CNRS
26. Vatsana Tiwari, Indian Institute of Science Education and Research, Bhopal, India
27. Florian Haberberger, LMU Munich
28. Nikolai Leopold, University of Basel
29. Leonard Wetzel, LMU Munich
30. Marcin Napiórkowski, University of Warsaw
31. Benjamin Schlein, University of Zurich
32. Robert Seiringer, IST Austria
33. Jan Philip Solovej, University of Copenhagen
34. Sabiha Tokus, Copenhagen University
35. Oleksandr Stashko, Taras Shevchenko National University of Kyiv
36. Oleh Savchuk, Taras Shevchenko National University in Kyiv
37. Mukhayo Rasulova, Institute of Nuclear Physics Uzbekistan Academy of Sciences
38. Slava Somsikov, Institute of Ionosphere
39. Théotime Girardot, LPMMC
40. Marcin Pruszczyk, University of Warsaw
41. Konstantin Merz, TU Braunschweig
42. DengLing Zhang, Xi’an University of Posts & Telecommunications
43. Christian Hainzl, LMU Munich
44. Ken Kim, LLNL/University of California
45. Aamna Ahmed, IISER Bhopal ,India
46. Michele Correggi, Politecnico di Milano
48. Peter Hislop, Mathematics Department, University of Kentucky
49. Andrea Moro, Università di Trento
50. Christiaan van de Ven, Università di Trento
51. Asbjørn Bækgaard Lauritsen, University of Copenhagen
52. Emanuela Giacomelli, LMU Münich
53. Soeren Petrat, Jacobs University Bremen
54. Giulia Basti, University of Zurich
55. Birger Brietzke, Heidelberg University
56. Alex Gottlieb, Wolfgang Pauli Institute
57. Søren Mikkelsen, University of Bristol
58. Johan Jacoby Klemmensen, Ludwig Maximilian University
59. Nilanjan Roy, Indian Institute of Science Education and Research Bhopal
60. Luca Fresta, University of Zurich
61. Andreas Deuchert, Institute of Mathematics, University of Zurich
62. Davide Fermi, Scuola Normale Superiore
63. Per Moosavi, ETH Zurich
64. Alessandro Olgiati, CNRS Grenoble
65. Chokri Manai, TU München
66. Julien Ricaud, LMU (Munich)
67. Marie Fialova, UCPH, QMATH
68. Cristina Caraci, Gran Sasso Science Institute
69. Johannes Agerskov, University of Copenhagen
70. Luca Dellantonio, Institute for Quantum Computing
71. Amanda Young, Technical University of Munich
72. Amirali Hannani, Universite Paris Dauphine-PSL
73. Igor Gapyak, Результати пошуку Результати веб-пошуку Taras Shevchenko National University of Kyiv
74. Lukas Schimmer, University of Copenhagen
75. Michał Pietruszka, University of Warsaw
76. Per von Soosten, Harvard University
77. Marcel Schaub, LMU Munich
78. Frederik Ravn Klausen, University of Copenhagen
79. Andrea Posilicano, Università dell'Insubria
80. Robert Rauch, TU Braunschweig, Germany
81. Li Chen, University of Mannheim
82. Marcello Porta, SISSA

18 further participants have registered.