Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in
L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others.
Full-text embargoed until: 2026-01-01Bose-Einstein condensation is a quantum statistical phase transition which was theoretically predicted almost a hundred years ago. After years of seminal research, physicists realized the first almost ideal Bose-Einstein condensates in ultracold dilute atomic gases in 1995. Since then, the theoretical and experimental methods concerning such systems have been developing rapidly, and many fascinating phenomena have been found in these novel quantum systems.\n\nBose-Einstein condensation occurs in a system consisting of massive bosons when a single quantum state becomes macroscopically occupied as the temperature is lowered below the transition temperature. In general, condensates consisting of repulsively interacting bosons exhibit superfluidity: Particle currents can flow in the system without dissipation and viscosity. Moreover, the velocity fields of condensates have to be irrotational, which severely restricts the rotational characteristics of these systems. Apart from the center of mass motion, the system may carry angular momentum in the form of elementary excitations or so-called quantized vortices.\n\nThis Thesis is a theoretical study of subjects related to stability and dynamics of quantized vortices in dilute atomic Bose-Einstein condensates. The precession and instability of off-centered vortices in trapped condensates is investigated both in the zero-temperature limit and at finite temperatures. Dynamical stability of multiply quantized vortices and vortex clusters is studied in axisymmetric trap geometries. Splitting of energetically and dynamically unstable multiply quantized vortices into singly quantized vortices is also studied. Finally, as a separate subject, tunneling of a condensate through a potential barrier is investigated. Majority of this work relies on numerical methods for solving the Gross-Pitaevskii and Bogoliubov equations, which are of central importance in the study of dilute atomic Bose-Einstein condensates.
The effective interaction energy of quasiparticles in the Bose fluid, which is described by ”ladder state diagrams” in the Bogoliubov approximation, was studied for a model potential. It is shown that in the Bogoliubov approximation, the interaction of excitations in liquid He can not be described even qualitatively. It means that a model of the weakly interacting dilute Bose gas fails in describing the interaction between excitation in Bose-condensed liquid, although this model qualitatively incorporates superfluidity in itself.