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Derivation of Hartree’s theory for mean-field Bose gases

Mathieu Lewin (2013)

Journées Équations aux dérivées partielles

This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of N bosons with an interaction of intensity 1 / N (mean-field regime). In the limit N → ∞ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation in these...

Des ponts

Cécile Dewitt-Morette (1998)

Publications Mathématiques de l'IHÉS

Diamagnetic behavior of sums Dirichlet eigenvalues

László Erdös, Michael Loss, Vitali Vougalter (2000)

Annales de l'institut Fourier

The Li-Yau semiclassical lower bound for the sum of the first N eigenvalues of the Dirichlet–Laplacian is extended to Dirichlet– Laplacians with constant magnetic fields. Our method involves a new diamagnetic inequality for constant magnetic fields.

Differential calculus on almost commutative algebras and applications to the quantum hyperplane

Cătălin Ciupală (2005)

Archivum Mathematicum

In this paper we introduce a new class of differential graded algebras named DG ρ -algebras and present Lie operations on this kind of algebras. We give two examples: the algebra of forms and the algebra of noncommutative differential forms of a  ρ -algebra. Then we introduce linear connections on a  ρ -bimodule M over a  ρ -algebra  A and extend these connections to the space of forms from A to M . We apply these notions to the quantum hyperplane.

Differential geometry over the structure sheaf: a way to quantum physics

Fischer, Gerald (1998)

Proceedings of the 17th Winter School "Geometry and Physics"

An idea for quantization by means of geometric observables is explained, which is a kind of the sheaf theoretical methods. First the formulation of differential geometry by using the structure sheaf is explained. The point of view to get interesting noncommutative observable algebras of geometric fields is introduced. The idea is to deform the algebra C ∞ ( M , ℝ ) by suitable interaction structures. As an example of such structures the Poisson-structure is mentioned and this leads naturally to deformation...

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