29,213 research outputs found
Second order splitting of a class of fourth order PDEs with point constraints
We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments
Ferromagnetism of the Hubbard Model at Strong Coupling in the Hartree-Fock Approximation
As a contribution to the study of Hartree-Fock theory we prove rigorously
that the Hartree-Fock approximation to the ground state of the d-dimensional
Hubbard model leads to saturated ferromagnetism when the particle density (more
precisely, the chemical potential mu) is small and the coupling constant U is
large, but finite. This ferromagnetism contradicts the known fact that there is
no magnetization at low density, for any U, and thus shows that HF theory is
wrong in this case. As in the usual Hartree-Fock theory we restrict attention
to Slater determinants that are eigenvectors of the z-component of the total
spin, {S}_z = sum_x n_{x,\uparrow} - n_{x,\downarrow}, and we find that the
choice 2{S}_z = N = particle number gives the lowest energy at fixed 0 < mu <
4d.Comment: v2: Published version. 30 pages latex. Changes in title, abstract,
introductio
Formation of ultracold LiRb molecules by photoassociation near the Li (2s 2S1/2) + Rb (5p 2P1/2) asymptote
We report the production of ultracold 7Li85Rb molecules by photoassociation
(PA) below the Li (2s 2S1/2) + Rb (5p 2P1/2) asymptote. We perform PA
spectroscopy in a dual-species 7Li-85Rb magneto-optical trap (MOT) and detect
the PA resonances using trap loss spectroscopy. We observe several strong PA
resonances corresponding to the last few bound states, assign the lines and
derive the long range C6 dispersion coefficients for the Li (2s 2S1/2) + Rb (5p
2P1/2) asymptote. We also report an excited-state molecule formation rate
(P_LiRb) of ~10^7 s^-1 and a PA rate coefficient (K_PA) of ~4x10^-11 cm^3/s,
which are both among the highest observed for heteronuclear bi-alkali
molecules. These suggest that PA is a promising route for the creation of
ultracold ground state LiRb molecules.Comment: 6 page
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