TY - JOUR
AB - We review recent progress towards a rigorous understanding of the Bogoliubov approximation for bosonic quantum many-body systems. We focus, in particular, on the excitation spectrum of a Bose gas in the mean-field (Hartree) limit. A list of open problems will be discussed at the end.
AU - Seiringer, Robert
ID - 1821
IS - 7
JF - Journal of Mathematical Physics
TI - Bose gases, Bose-Einstein condensation, and the Bogoliubov approximation
VL - 55
ER -
TY - JOUR
AU - Jakšić, Vojkan
AU - Pillet, Claude
AU - Seiringer, Robert
ID - 1822
IS - 7
JF - Journal of Mathematical Physics
TI - Introduction
VL - 55
ER -
TY - JOUR
AB - 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 also includes 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.
AU - Bräunlich, Gerhard
AU - Hainzl, Christian
AU - Seiringer, Robert
ID - 1889
IS - 7
JF - Reviews in Mathematical Physics
TI - Translation-invariant quasi-free states for fermionic systems and the BCS approximation
VL - 26
ER -
TY - JOUR
AB - We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation with a time-dependent potential and we show the existence of the wave operator in Schatten spaces.
AU - Frank, Rupert
AU - Lewin, Mathieu
AU - Lieb, Élliott
AU - Seiringer, Robert
ID - 1904
IS - 7
JF - Journal of the European Mathematical Society
TI - Strichartz inequality for orthonormal functions
VL - 16
ER -
TY - JOUR
AB - As the nuclear charge Z is continuously decreased an N-electron atom undergoes a binding-unbinding transition. We investigate whether the electrons remain bound and whether the radius of the system stays finite as the critical value Zc is approached. Existence of a ground state at Zc is shown under the condition Zc < N-K, where K is the maximal number of electrons that can be removed at Zc without changing the energy.
AU - Bellazzini, Jacopo
AU - Frank, Rupert
AU - Lieb, Élliott
AU - Seiringer, Robert
ID - 1918
IS - 1
JF - Reviews in Mathematical Physics
TI - Existence of ground states for negative ions at the binding threshold
VL - 26
ER -
TY - JOUR
AB - We consider Ising models in d = 2 and d = 3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)-p, p > 2d, at large distances. If the strength J of the ferromagnetic interaction is larger than a critical value J c, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to J c, the ground state is periodic and striped, with stripes of constant width h = h(J), and h → ∞ as J → Jc -. (In d = 3 stripes mean slabs, not columns.) Here we rigorously prove that, if we normalize the energy in such a way that the energy of the homogeneous state is zero, then the ratio e 0(J)/e S(J) tends to 1 as J → Jc -, with e S(J) being the energy per site of the optimal periodic striped/slabbed state and e 0(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e 0(J)-e S(J) at small but positive J c-J, and also shows that in this parameter range the ground state is striped/slabbed in a certain sense: namely, if one looks at a randomly chosen window, of suitable size ℓ (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability.
AU - Giuliani, Alessandro
AU - Lieb, Élliott
AU - Seiringer, Robert
ID - 1935
IS - 1
JF - Communications in Mathematical Physics
TI - Formation of stripes and slabs near the ferromagnetic transition
VL - 331
ER -
TY - JOUR
AB - Spin-wave theory is a key ingredient in our comprehension of quantum spin systems, and is used successfully for understanding a wide range of magnetic phenomena, including magnon condensation and stability of patterns in dipolar systems. Nevertheless, several decades of research failed to establish the validity of spin-wave theory rigorously, even for the simplest models of quantum spins. A rigorous justification of the method for the three-dimensional quantum Heisenberg ferromagnet at low temperatures is presented here. We derive sharp bounds on its free energy by combining a bosonic formulation of the model introduced by Holstein and Primakoff with probabilistic estimates and operator inequalities.
AU - Correggi, Michele
AU - Giuliani, Alessandro
AU - Seiringer, Robert
ID - 2029
IS - 2
JF - EPL
TI - Validity of spin-wave theory for the quantum Heisenberg model
VL - 108
ER -
TY - JOUR
AB - We prove the existence of scattering states for the defocusing cubic Gross-Pitaevskii (GP) hierarchy in ℝ3. Moreover, we show that an exponential energy growth condition commonly used in the well-posedness theory of the GP hierarchy is, in a specific sense, necessary. In fact, we prove that without the latter, there exist initial data for the focusing cubic GP hierarchy for which instantaneous blowup occurs.
AU - Chen, Thomas
AU - Hainzl, Christian
AU - Pavlović, Nataša
AU - Seiringer, Robert
ID - 2186
IS - 7
JF - Letters in Mathematical Physics
TI - On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti
VL - 104
ER -
TY - CONF
AB - 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.
AU - Bräunlich, Gerhard
AU - Hainzl, Christian
AU - Seiringer, Robert
ID - 1516
T2 - Proceedings of the QMath12 Conference
TI - On the BCS gap equation for superfluid fermionic gases
ER -
TY - JOUR
AB - We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The presentation includes a discussion of Bose-Einstein condensation, the excitation spectrum for trapped gases and its relation to superfluidity, as well as the appearance of quantized vortices in rotating systems. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schrödinger equation.
AU - Seiringer, Robert
ID - 2297
IS - 2
JF - Japanese Journal of Mathematics
TI - Hot topics in cold gases: A mathematical physics perspective
VL - 8
ER -
TY - JOUR
AB - We consider Ising models in two and three dimensions with nearest neighbor ferromagnetic interactions and long-range, power law decaying, antiferromagnetic interactions. If the strength of the ferromagnetic coupling J is larger than a critical value Jc, then the ground state is homogeneous and ferromagnetic. As the critical value is approached from smaller values of J, it is believed that the ground state consists of a periodic array of stripes (d=2) or slabs (d=3), all of the same size and alternating magnetization. Here we prove rigorously that the ground state energy per site converges to that of the optimal periodic striped or slabbed state, in the limit that J tends to the ferromagnetic transition point. While this theorem does not prove rigorously that the ground state is precisely striped or slabbed, it does prove that in any suitably large box the ground state is striped or slabbed with high probability.
AU - Giuliani, Alessandro
AU - Lieb, Élliott
AU - Seiringer, Robert
ID - 2300
IS - 6
JF - Physical Review B
TI - Realization of stripes and slabs in two and three dimensions
VL - 88
ER -
TY - JOUR
AB - We show that bosons interacting via pair potentials with negative scattering length form bound states for a suitable number of particles. In other words, the absence of many-particle bound states of any kind implies the non-negativity of the scattering length of the interaction potential.
AU - Seiringer, Robert
ID - 2318
IS - 3
JF - Journal of Spectral Theory
TI - Absence of bound states implies non-negativity of the scattering length
VL - 2
ER -