@article{2374,
abstract = {A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density Q and temperature T. In the dilute regime, i.e., when a3 1, where a denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term 4πa(2 2}-[ - c]2+). Here, c(T) denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and [ · ]+ = max{ ·, 0} denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., T ~ 2/3 or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [17] for estimating correlations to temperatures below the critical one.},
author = {Robert Seiringer},
journal = {Communications in Mathematical Physics},
number = {3},
pages = {595 -- 636},
publisher = {Springer},
title = {{Free energy of a dilute Bose gas: Lower bound}},
doi = {10.1007/s00220-008-0428-2},
volume = {279},
year = {2008},
}
@article{2376,
abstract = {We derive upper and lower bounds on the critical temperature Tc and the energy gap Ξ (at zero temperature) for the BCS gap equation, describing spin- 1 2 fermions interacting via a local two-body interaction potential λV(x). At weak coupling λ 1 and under appropriate assumptions on V(x), our bounds show that Tc ∼A exp(-B/λ) and Ξ∼C exp(-B/λ) for some explicit coefficients A, B, and C depending on the interaction V(x) and the chemical potential μ. The ratio A/C turns out to be a universal constant, independent of both V(x) and μ. Our analysis is valid for any μ; for small μ, or low density, our formulas reduce to well-known expressions involving the scattering length of V(x).},
author = {Hainzl, Christian and Robert Seiringer},
journal = {Physical Review B - Condensed Matter and Materials Physics},
number = {18},
publisher = {American Physical Society},
title = {{Critical temperature and energy gap for the BCS equation}},
doi = {10.1103/PhysRevB.77.184517},
volume = {77},
year = {2008},
}
@article{2377,
abstract = {We prove that the critical temperature for the BCS gap equation is given by T c = μ ( 8\π e γ-2+ o(1)) e π/(2μa) in the low density limit μ→ 0, with γ denoting Euler's constant. The formula holds for a suitable class of interaction potentials with negative scattering length a in the absence of bound states.},
author = {Hainzl, Christian and Robert Seiringer},
journal = {Letters in Mathematical Physics},
number = {2-3},
pages = {99 -- 107},
publisher = {Springer},
title = {{The BCS critical temperature for potentials with negative scattering length}},
doi = {10.1007/s11005-008-0242-y},
volume = {84},
year = {2008},
}
@article{2378,
abstract = {We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani (J. Math. Phys. 48:023302, [2007]), our result proves that in the low density limit the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by 8πaσ uσ d , where σ u(d) denotes the density of the spin-up (down) particles, and a is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.},
author = {Robert Seiringer and Yin, Jun},
journal = {Journal of Statistical Physics},
number = {6},
pages = {1139 -- 1154},
publisher = {Springer},
title = {{Ground state energy of the low density hubbard model}},
doi = {10.1007/s10955-008-9527-x},
volume = {131},
year = {2008},
}
@article{2379,
author = {Frank, Rupert L and Lieb, Élliott H and Robert Seiringer},
journal = {Journal of the American Mathematical Society},
number = {4},
pages = {925 -- 950},
publisher = {American Mathematical Society},
title = {{Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators}},
doi = {10.1090/S0894-0347-07-00582-6},
volume = {21},
year = {2008},
}
@article{2380,
abstract = {The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.},
author = {Hainzl, Christian and Hamza, Eman and Robert Seiringer and Solovej, Jan P},
journal = {Communications in Mathematical Physics},
number = {2},
pages = {349 -- 367},
publisher = {Springer},
title = {{The BCS functional for general pair interactions}},
doi = {10.1007/s00220-008-0489-2},
volume = {281},
year = {2008},
}
@article{2381,
abstract = {We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.},
author = {Frank, Rupert L and Robert Seiringer},
journal = {Journal of Functional Analysis},
number = {12},
pages = {3407 -- 3430},
publisher = {Academic Press},
title = {{Non-linear ground state representations and sharp Hardy inequalities}},
doi = {10.1016/j.jfa.2008.05.015},
volume = {255},
year = {2008},
}
@article{2382,
abstract = {We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δ-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length a and the radius r of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant g ∼ a/r 2 is derived. Our bounds are uniform in g in the whole parameter range 0 ≤ g ≤ ∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ∼ r -2 above the ground state energy.},
author = {Robert Seiringer and Yin, Jun},
journal = {Communications in Mathematical Physics},
number = {2},
pages = {459 -- 479},
publisher = {Springer},
title = {{The Lieb-Liniger model as a limit of dilute bosons in three dimensions}},
doi = {10.1007/s00220-008-0521-6},
volume = {284},
year = {2008},
}
@article{2383,
abstract = {We study the relativistic electron-positron field at positive temperature in the Hartree-Fock approximation. We consider both the case with and without exchange terms, and investigate the existence and properties of minimizers. Our approach is non-perturbative in the sense that the relevant electron subspace is determined in a self-consistent way. The present work is an extension of previous work by Hainzl, Lewin, Séré and Solovej where the case of zero temperature was considered.},
author = {Hainzl, Christian and Lewin, Mathieu and Robert Seiringer},
journal = {Reviews in Mathematical Physics},
number = {10},
pages = {1283 -- 1307},
publisher = {World Scientific Publishing},
title = {{A nonlinear model for relativistic electrons at positive temperature}},
doi = {10.1142/S0129055X08003547},
volume = {20},
year = {2008},
}
@inproceedings{2702,
abstract = {We review our proof that in a scaling limit, the time evolution of a quantum particle in a static random environment leads to a diffusion equation. In particular, we discuss the role of Feynman graph expansions and of renormalization.
},
author = {László Erdös and Salmhofer, Manfred and Yau, Horng-Tzer},
pages = {167 -- 182},
publisher = {World Scientific Publishing},
title = {{Feynman graphs and renormalization in quantum diffusion}},
doi = {10.1142/9789812833556_0011},
year = {2008},
}