arXiv subjects
Publications and source records attributed to Christian Brennecke.
We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $N^{-1+κ}$, for $κ>0$. Assuming that $κ\in (0;1/43)$, we show that low-energy states of the system exhibit complete Bose-Einstein condensation by providing explicit bounds on the expectation and on higher moments of the number of excitations.
We consider the quantum Sherrington-Kirkpatrick (SK) spin-glass model with transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures $β>0$. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with $\mathbb{R}^d$-valued spins. This enables us to use results of Panchenko who generalized in \cite{Pan2,Pan3} the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynamic limit of the free energy of the quantum SK model as the $d\to\infty$ limit of the free energies of the $d$-dimensional approximations of the model.
We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in \cite{BBCS1}, removing the assumption of small interaction potential.
We consider Bose gases consisting of $N$ particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of the order $N^{-1}$(Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as $N \to \infty$. Our results confirm Bogoliubov's predictions.
We consider a system of $N$ bosons interacting through a singular two-body potential scaling with $N$ and having the form $N^{3β-1} V (N^βx)$, for an arbitrary parameter $β\in (0,1)$. We provide a norm-approximation for the many-body evolution of initial data exhibiting Bose-Einstein condensation in terms of a cubic nonlinear Schrödinger equation for the condensate wave function and of a unitary Fock space evolution with a generator quadratic in creation and annihilation operators for the fluctuations.
We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for $N$-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.
We consider systems of $N$ bosons in a box with volume one, interacting through a repulsive two-body potential of the form $κN^{3β-1} V(N^βx)$. For all $0 < β< 1$, and for sufficiently small coupling constant $κ> 0$, we establish the validity of Bogoliubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large $N$.