We consider the Bogolubov-Hartree-Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose-Einstein condensate of fermion pairs. The latter can be described by means of the Gross-Pitaevskii energy functional. * c 2015 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes. arXiv:1511.08047v1 [math-ph] 25 Nov 2015 1 2h . This assumes, of course, that the two-body interaction potential V allows for a negative energy bound state, which is part of the following assumption.and −2∆ + V has a normalized ground state α 0 with corresponding ground state energy −E b < 0.
We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but includes also direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity. *
We consider expansions of eigenvalues and eigenvectors of models of quantum field theory. For a class of models known as generalized spin boson model we prove the existence of asymptotic expansions of the ground state and the ground state energy to arbitrary order. We need a mild but very natural infrared assumption, which is weaker than the assumption usually needed for other methods such as operator theoretic renormalization to be applicable. The result complements previously shown analyticity properties.
We present a rigorous derivation of the BCS gap equation for superfluid fermionic gases with point interactions. Our starting point is the BCS energy functional, whose minimizer we investigate in the limit when the range of the interaction potential goes to zero.
Based on the geodesic equation in a static spherically symmetric metric we discuss the rotation curve and gravitational lensing. The rotation curve determines one function in the metric without assuming Einstein's equations. Then lensing is considered in the weak field approximation of general relativity. From the null geodesics we derive the lensing equation. The gravitational potential U (r ) which determines the lensing is directly give by the rotation curve U (r ) = −v 2 (r ). This allows to test general relativity on the scale of galaxies where dark matter is relevant.
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