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arXiv:2309.12233

Ground state of Bose gases interacting through singular potentials

Lea Boßmann, Nikolai Leopold, Sören Petrat, Simone Rademacher

Published Sep 21, 2023• 0 citations
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Abstract

We consider a system of $N$ bosons on the three-dimensional unit torus. The particles interact through repulsive pair interactions of the form $N^{3β-1} v (N^βx)$ for $β\in (0,1)$. We prove the next order correction to Bogoliubov theory for the ground state and the ground state energy.

DOI
10.48550/arxiv.2309.12233

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We consider a system of NNN bosons on the three-dimensional unit torus. The particles interact through repulsive pair interactions of the form N3β−1v(Nβx)N^{3β-1} v (N^βx)N3β−1v(Nβx) for β∈(0,1)β\in (0,1)β∈(0,1). We prove the next order correction to Bogoliubov theory for the ground state and the ground state energy.
Abstract We consider a system of N bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in N (Gross–Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit N→∞N \rightarrow \infty N→∞ N → ∞ .
The isotropic scattering phase shift is calculated for nonrelativistic bosons interacting at low energies via an arbitrary finite-range potential in ddd space-time dimensions. Scattering on a $(d\ensuremath{-}1)$-dimensional torus is then considered, and the eigenvalue equation relating the energy levels on the torus to the scattering phase shift is derived. With this technology in hand, and focusing on the case of two spatial dimensions, a perturbative expansion is developed for the ground-state energy of NNN identical bosons which interact via an arbitrary finite-range potential in a finite area. The leading nonuniversal effects due to range corrections and three-body forces are included. It is then shown that the thermodynamic limit of the ground-state energy in a finite area can be taken in closed form to obtain the energy per particle in the low-density expansion by explicitly summing the parts of the finite-area energy that diverge with powers of NNN. The leading and subleading finite-size corrections to the thermodynamic limit equation of state are also computed. Closed-form results---some well known, others perhaps not---for two-dimensional lattice sums are included in an Appendix.
We study the large‐N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and nondegenerate, and that there is complete Bose‐Einstein condensation on this state. Using our method we then treat two applications: atoms with “bosonic” electrons on one hand, and trapped two‐dimensional and three‐dimensional Coulomb gases on the other hand. © 2015 Wiley Periodicals, Inc.