Periodic Reporting for period 4 - CLaQS (Correlations in Large Quantum Systems)
Reporting period: 2024-03-01 to 2025-02-28
This project is devoted to the mathematical analysis of important physical properties of many-body quantum systems. We are interested in properties of the ground state and low-energy excitations but also of non-equilibrium dynamics. We consider systems with different statistics and in different regimes. The questions we are addressing have a common aspect: correlations among particles play a crucial role. Our main goal consists in developing new tools that allow us to correctly describe many-body correlations and to understand their effects.
The starting point of the project are ideas and techniques that have been introduced in the last years, establishing the validity of Bogoliubov theory for Bose gases in the Gross-Pitaevskii regime and showing that (bosonic) Bogoliubov theory can also be used to study the correlation energy of Fermi gases. The overall objective of the project consists in further developing these new techniques and in applying them to new contexts. We believe they have the potential to approach some fundamental open questions in mathematical physics. Specifically, the goals of the project, as initially stated in the proposal are as follows: 1) Excitation spectrum for Bose gases trapped by external potential, 2) Norm approximation for dynamics in GP regime, 3) Bogoliubov excitation spectrum for large interaction potential, 4) Lee-Huang-Yang formula for dilute Bose gases, 5) BEC in the thermodynamic limit, 6) Gell Mann--Brueckner formula for mean-field fermions, 7) Huang-Yang formula for dilute fermions, 8) Derivation of quantum Boltzmann equation.
The starting point of the project are ideas and techniques that have been introduced in the last years, establishing the validity of Bogoliubov theory for Bose gases in the Gross-Pitaevskii regime and showing that (bosonic) Bogoliubov theory can also be used to study the correlation energy of Fermi gases. The overall objective of the project consists in further developing these new techniques and in applying them to new contexts. We believe they have the potential to approach some fundamental open questions in mathematical physics. Specifically, the goals of the project, as initially stated in the proposal are as follows: 1) Excitation spectrum for Bose gases trapped by external potential, 2) Norm approximation for dynamics in GP regime, 3) Bogoliubov excitation spectrum for large interaction potential, 4) Lee-Huang-Yang formula for dilute Bose gases, 5) BEC in the thermodynamic limit, 6) Gell Mann--Brueckner formula for mean-field fermions, 7) Huang-Yang formula for dilute fermions, 8) Derivation of quantum Boltzmann equation.
During the project, we focussed mainly on the objectives 1),2),3),4),5),6). Let us describe the most important progress that we achieved.
Objective 1: Bogoliubov theory for trapped. This objective has been reached through the two papers [1,2], where we established the validity of Bogoliubov theory for Bose gases trapped by generic external potentials, providing precise estimated for the ground state energy of the gas and for its low-lying excitation spectrum.
Objective 2): Norm approximation for many-body dynamics. This goal has been recently reached in [3]. In this paper, we managed to provide a norm-approximation for the many-body evolution of a Bose gas in the Gross-Pitaevskii regime. To approximate the solution of the many-body Schrödinger equation, we combine the nonlinear time-dependent Gross-Pitaevskii equation, describing the dynamics of the Bose-Einstein condensate, with a time-dependent Bogoliubov transformation, describing the evolution of the excitations of the condensate.
Objective 3): Bogoliubov theory for large interactions. The validity of Bogoliubov theory for Bose gases in the Gross-Piteavskii regime, with no assumption on the size of the potential, was achieved in [4]. During the funding period, we continued to work in this direction. In [5], we managed to handle a Bose gas in the Gross-Pitaevskii regime, interacting through a hard sphere potential. This particularly interesting and natural choice of the interaction potential was excluded in [4]. In [6,7], we proposed new approaches to show the validity of Bogoliubov theory in the Gross-Pitaevskii regime, simplifying substantially the proof of [4].
Objective 4): Lee-Huang-Yang formula. In [8] we derived a new upper bound for the ground state energy of a dilute Bose gas, matching the prediction of Lee-Huang-Yang (substantially extending and simplifying previous results). While the results of [8] are restricted to integrable interaction potentials, in [9] we recently obtained an upper bound for hard spheres; this bound reaches the precision of the Lee-Huang-Yang estimate, but fails to reproduce the correct constant. It remains an open problem to establish the validity of the Lee-Huang-Yang conjecture for hard sphere interactions.
Objective 5): Bose-Einstein condensation in the thermodynamic limit. We managed in [10,11] to extend previous results on Bose-Einstein condensation and on the validity of Bogoliubov theory to scaling regimes interpolating between the Gross-Pitaevskii and the thermodynamic limit. So far, we are still far from the most interesting but also most challenging thermodynamic limit. However, there is some hope that the simplified approaches developed in [6,7] may be useful to make progress in this direction.
Objective 6): Correlation energy for Fermi gases. In [12] we were able to obtain rigorous derivations of the Gell-Mann - Brueckner formula for the correlation energy of Fermi gases in the mean-field regime (which, for fermions, is naturally linked with a semiclassical limit). Recently, we turned our attention to Fermi gases at high density, in the thermodynamic limit. In [13] we considered the time-evolution of extended Fermi gases and we proved that it can be approximated by the nonlinear Hartree-Fock dynamics.
[1] Brennecke, Schlein, Schraven. Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime. Math. Phys. An. Geom. 25 (2022).
[2] Brennecke, Schlein, Schraven. Bogoliubov Theory for Trapped Bosons in the Gross-Pitaevskii Regime. Ann. Henri Poincare' 23 (2022).
[3] Caraci, Oldenburg, Schlein. Quantum fluctuations of many-body dynamics around the Gross–Pitaevskii equation. Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire (2024).
[4] Boccato, Brennecke, Cenatiempo, Schlein. Bogoliubov Theory in the Gross-Pitaevskii Limit. Acta Math. 222 (2019).
[5] Basti, Cenatiempo, Olgiati, Pasqualetti, Schlein. A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime. Comm. Math. Phys. (2022).
[6] Hainzl, Schlein, Triay. Bogoliubov Theory in the Gross-Pitaevskii Limit: a Simplified Approach. Forum of Math. Sigma 10 (2022).
[7] Brooks. Diagonalizing Bose Gases in the Gross-Pitaevskii Regime and Beyond. Commun. Math. Phys. (2025).
[8] Basti, Cenatiempo, Schlein. A new second order upper bound for the ground state energy of dilute Bose gases. Forum of Math., Sigma 9 (2021).
[9] Basti, Cenatiempo, Giuliani, Pasqualetti, Olgiati, Schlein. Upper bound for the ground state energy of a dilute Bose gas of hard spheres. Arch. Rational Mech. Anal. (2024).
[10] Adhikari, Brennecke, Schlein. Bose-Einstein condensation beyond the Gross-Pitaevskii regime. Ann. H. Poincare' 22 (2021).
[11] Brennecke, Caporaletti, Schlein. Excitation Spectrum for Bose Gases beyond the Gross-Pitaevskii Regime. Rev. Math. Phys. 34 (2022).
[12] Benedikter, Nam, Porta, Schlein, Seiringer. Correlation energy of a weakly interacting Fermi gas. Inventiones Math. 225 (2021).
[13] Fresta, Porta, Schlein. Effective Dynamics of Extended Fermi Gases in the High-Density Regime. Commun. Math. Phys. 401 (2023).
Objective 1: Bogoliubov theory for trapped. This objective has been reached through the two papers [1,2], where we established the validity of Bogoliubov theory for Bose gases trapped by generic external potentials, providing precise estimated for the ground state energy of the gas and for its low-lying excitation spectrum.
Objective 2): Norm approximation for many-body dynamics. This goal has been recently reached in [3]. In this paper, we managed to provide a norm-approximation for the many-body evolution of a Bose gas in the Gross-Pitaevskii regime. To approximate the solution of the many-body Schrödinger equation, we combine the nonlinear time-dependent Gross-Pitaevskii equation, describing the dynamics of the Bose-Einstein condensate, with a time-dependent Bogoliubov transformation, describing the evolution of the excitations of the condensate.
Objective 3): Bogoliubov theory for large interactions. The validity of Bogoliubov theory for Bose gases in the Gross-Piteavskii regime, with no assumption on the size of the potential, was achieved in [4]. During the funding period, we continued to work in this direction. In [5], we managed to handle a Bose gas in the Gross-Pitaevskii regime, interacting through a hard sphere potential. This particularly interesting and natural choice of the interaction potential was excluded in [4]. In [6,7], we proposed new approaches to show the validity of Bogoliubov theory in the Gross-Pitaevskii regime, simplifying substantially the proof of [4].
Objective 4): Lee-Huang-Yang formula. In [8] we derived a new upper bound for the ground state energy of a dilute Bose gas, matching the prediction of Lee-Huang-Yang (substantially extending and simplifying previous results). While the results of [8] are restricted to integrable interaction potentials, in [9] we recently obtained an upper bound for hard spheres; this bound reaches the precision of the Lee-Huang-Yang estimate, but fails to reproduce the correct constant. It remains an open problem to establish the validity of the Lee-Huang-Yang conjecture for hard sphere interactions.
Objective 5): Bose-Einstein condensation in the thermodynamic limit. We managed in [10,11] to extend previous results on Bose-Einstein condensation and on the validity of Bogoliubov theory to scaling regimes interpolating between the Gross-Pitaevskii and the thermodynamic limit. So far, we are still far from the most interesting but also most challenging thermodynamic limit. However, there is some hope that the simplified approaches developed in [6,7] may be useful to make progress in this direction.
Objective 6): Correlation energy for Fermi gases. In [12] we were able to obtain rigorous derivations of the Gell-Mann - Brueckner formula for the correlation energy of Fermi gases in the mean-field regime (which, for fermions, is naturally linked with a semiclassical limit). Recently, we turned our attention to Fermi gases at high density, in the thermodynamic limit. In [13] we considered the time-evolution of extended Fermi gases and we proved that it can be approximated by the nonlinear Hartree-Fock dynamics.
[1] Brennecke, Schlein, Schraven. Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime. Math. Phys. An. Geom. 25 (2022).
[2] Brennecke, Schlein, Schraven. Bogoliubov Theory for Trapped Bosons in the Gross-Pitaevskii Regime. Ann. Henri Poincare' 23 (2022).
[3] Caraci, Oldenburg, Schlein. Quantum fluctuations of many-body dynamics around the Gross–Pitaevskii equation. Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire (2024).
[4] Boccato, Brennecke, Cenatiempo, Schlein. Bogoliubov Theory in the Gross-Pitaevskii Limit. Acta Math. 222 (2019).
[5] Basti, Cenatiempo, Olgiati, Pasqualetti, Schlein. A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime. Comm. Math. Phys. (2022).
[6] Hainzl, Schlein, Triay. Bogoliubov Theory in the Gross-Pitaevskii Limit: a Simplified Approach. Forum of Math. Sigma 10 (2022).
[7] Brooks. Diagonalizing Bose Gases in the Gross-Pitaevskii Regime and Beyond. Commun. Math. Phys. (2025).
[8] Basti, Cenatiempo, Schlein. A new second order upper bound for the ground state energy of dilute Bose gases. Forum of Math., Sigma 9 (2021).
[9] Basti, Cenatiempo, Giuliani, Pasqualetti, Olgiati, Schlein. Upper bound for the ground state energy of a dilute Bose gas of hard spheres. Arch. Rational Mech. Anal. (2024).
[10] Adhikari, Brennecke, Schlein. Bose-Einstein condensation beyond the Gross-Pitaevskii regime. Ann. H. Poincare' 22 (2021).
[11] Brennecke, Caporaletti, Schlein. Excitation Spectrum for Bose Gases beyond the Gross-Pitaevskii Regime. Rev. Math. Phys. 34 (2022).
[12] Benedikter, Nam, Porta, Schlein, Seiringer. Correlation energy of a weakly interacting Fermi gas. Inventiones Math. 225 (2021).
[13] Fresta, Porta, Schlein. Effective Dynamics of Extended Fermi Gases in the High-Density Regime. Commun. Math. Phys. 401 (2023).
The progress beyond the state of the art has already been explained in the previous answer. Until the end of the project, we expect to make additional progress in the directions indicated in the proposal and, possibly, to find satisfactory answers to some of the questions left open. Moreover, we expect that (as it has been the case so far) the techniques developed to solve the problems listed in the proposal will allow us to answer questions in other contexts, as well.