@article{473, abstract = {We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature T diverges and the interaction strength behaves as 1/T. We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dimensional analogue of phase-space semiclassical analysis, using fine properties of the quantum relative entropy, the link between quantum de Finetti measures and upper/lower symbols in a coherent state basis, as well as Berezin-Lieb type inequalities. Our results cover the measure built on the defocusing nonlinear Schrödinger functional on a finite interval, as well as smoother interactions in dimensions d 2.}, author = {Lewin, Mathieu and Phan Thanh, Nam and Rougerie, Nicolas}, journal = {Journal de l'Ecole Polytechnique - Mathematiques}, pages = {65 -- 115}, publisher = {Ecole Polytechnique}, title = {{Derivation of nonlinear gibbs measures from many-body quantum mechanics}}, doi = {10.5802/jep.18}, volume = {2}, year = {2015}, } @article{1572, abstract = {We consider the quantum ferromagnetic Heisenberg model in three dimensions, for all spins S ≥ 1/2. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy at low temperatures. Our proof comes with explicit, constructive upper and lower bounds on the error term. It uses in an essential way the bosonic formulation of the model in terms of the Holstein-Primakoff representation. In this language, the model describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor attraction. This attractive interaction makes the lower bound on the free energy particularly tricky: the key idea there is to prove a differential inequality for the two-particle density, which is thereby shown to be smaller than the probability density of a suitably weighted two-particle random process on the lattice. }, author = {Correggi, Michele and Giuliani, Alessandro and Seiringer, Robert}, journal = {Communications in Mathematical Physics}, number = {1}, pages = {279 -- 307}, publisher = {Springer}, title = {{Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet}}, doi = {10.1007/s00220-015-2402-0}, volume = {339}, year = {2015}, } @article{1573, abstract = {We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in ℝ3. One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdos, Schlein, and Yau.}, author = {Chen, Thomas and Hainzl, Christian and Pavlović, Nataša and Seiringer, Robert}, journal = {Communications on Pure and Applied Mathematics}, number = {10}, pages = {1845 -- 1884}, publisher = {Wiley}, title = {{Unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti}}, doi = {10.1002/cpa.21552}, volume = {68}, year = {2015}, } @article{10814, abstract = {We review recent progress towards a rigorous understanding of the excitation spectrum of bosonic quantum many-body systems. In particular, we explain how one can rigorously establish the predictions resulting from the Bogoliubov approximation in the mean field limit. The latter predicts that the spectrum is made up of elementary excitations, whose energy behaves linearly in the momentum for small momentum. This property is crucial for the superfluid behavior of the system. We also discuss a list of open problems in this field.}, author = {Seiringer, Robert}, issn = {1869-7135}, journal = {Jahresbericht der Deutschen Mathematiker-Vereinigung}, keywords = {General Medicine}, pages = {21--41}, publisher = {Springer Nature}, title = {{The excitation spectrum for Bose fluids with weak interactions}}, doi = {10.1365/s13291-014-0083-9}, volume = {116}, year = {2014}, } @inproceedings{8044, abstract = {Many questions concerning models in quantum mechanics require a detailed analysis of the spectrum of the corresponding Hamiltonian, a linear operator on a suitable Hilbert space. Of particular relevance for an understanding of the low-temperature properties of a system is the structure of the excitation spectrum, which is the part of the spectrum close to the spectral bottom. We present recent progress on this question for bosonic many-body quantum systems with weak two-body interactions. Such system are currently of great interest, due to their experimental realization in ultra-cold atomic gases. We investigate the accuracy of the Bogoliubov approximations, which predicts that the low-energy spectrum is made up of sums of elementary excitations, with linear dispersion law at low momentum. The latter property is crucial for the superfluid behavior the system.}, author = {Seiringer, Robert}, booktitle = {Proceeding of the International Congress of Mathematicans}, isbn = {9788961058063}, location = {Seoul, South Korea}, pages = {1175--1194}, publisher = {International Congress of Mathematicians}, title = {{Structure of the excitation spectrum for many-body quantum systems}}, volume = {3}, year = {2014}, } @article{1821, abstract = {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.}, author = {Seiringer, Robert}, journal = {Journal of Mathematical Physics}, number = {7}, publisher = {American Institute of Physics}, title = {{Bose gases, Bose-Einstein condensation, and the Bogoliubov approximation}}, doi = {10.1063/1.4881536}, volume = {55}, year = {2014}, } @article{1822, author = {Jakšić, Vojkan and Pillet, Claude and Seiringer, Robert}, journal = {Journal of Mathematical Physics}, number = {7}, publisher = {American Institute of Physics}, title = {{Introduction}}, doi = {10.1063/1.4884877}, volume = {55}, year = {2014}, } @article{1889, abstract = {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.}, author = {Bräunlich, Gerhard and Hainzl, Christian and Seiringer, Robert}, journal = {Reviews in Mathematical Physics}, number = {7}, publisher = {World Scientific Publishing}, title = {{Translation-invariant quasi-free states for fermionic systems and the BCS approximation}}, doi = {10.1142/S0129055X14500123}, volume = {26}, year = {2014}, } @article{1904, abstract = {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.}, author = {Frank, Rupert and Lewin, Mathieu and Lieb, Élliott and Seiringer, Robert}, journal = {Journal of the European Mathematical Society}, number = {7}, pages = {1507 -- 1526}, publisher = {European Mathematical Society}, title = {{Strichartz inequality for orthonormal functions}}, doi = {10.4171/JEMS/467}, volume = {16}, year = {2014}, } @article{1918, abstract = {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.}, author = {Bellazzini, Jacopo and Frank, Rupert and Lieb, Élliott and Seiringer, Robert}, journal = {Reviews in Mathematical Physics}, number = {1}, publisher = {World Scientific Publishing}, title = {{Existence of ground states for negative ions at the binding threshold}}, doi = {10.1142/S0129055X13500219}, volume = {26}, year = {2014}, } @article{1935, abstract = {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.}, author = {Giuliani, Alessandro and Lieb, Élliott and Seiringer, Robert}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {333 -- 350}, publisher = {Springer}, title = {{Formation of stripes and slabs near the ferromagnetic transition}}, doi = {10.1007/s00220-014-1923-2}, volume = {331}, year = {2014}, } @article{2029, abstract = {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.}, author = {Correggi, Michele and Giuliani, Alessandro and Seiringer, Robert}, journal = {EPL}, number = {2}, publisher = {IOP Publishing Ltd.}, title = {{Validity of spin-wave theory for the quantum Heisenberg model}}, doi = {10.1209/0295-5075/108/20003}, volume = {108}, year = {2014}, } @article{2186, abstract = {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.}, author = {Chen, Thomas and Hainzl, Christian and Pavlović, Nataša and Seiringer, Robert}, journal = {Letters in Mathematical Physics}, number = {7}, pages = {871 -- 891}, publisher = {Springer}, title = {{On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti}}, doi = {10.1007/s11005-014-0693-2}, volume = {104}, year = {2014}, } @article{2281, abstract = {We consider two-dimensional Bose-Einstein condensates with attractive interaction, described by the Gross-Pitaevskii functional. Minimizers of this functional exist only if the interaction strength a satisfies {Mathematical expression}, where Q is the unique positive radial solution of {Mathematical expression} in {Mathematical expression}. We present a detailed analysis of the behavior of minimizers as a approaches a*, where all the mass concentrates at a global minimum of the trapping potential.}, author = {Guo, Yujin and Seiringer, Robert}, journal = {Letters in Mathematical Physics}, number = {2}, pages = {141 -- 156}, publisher = {Springer}, title = {{On the mass concentration for Bose-Einstein condensates with attractive interactions}}, doi = {10.1007/s11005-013-0667-9}, volume = {104}, year = {2014}, } @inproceedings{1516, abstract = {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. }, author = {Bräunlich, Gerhard and Hainzl, Christian and Seiringer, Robert}, booktitle = {Proceedings of the QMath12 Conference}, location = {Berlin, Germany}, pages = {127 -- 137}, publisher = {World Scientific Publishing}, title = {{On the BCS gap equation for superfluid fermionic gases}}, doi = {10.1142/9789814618144_0007}, year = {2014}, } @article{2297, abstract = {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.}, author = {Seiringer, Robert}, journal = {Japanese Journal of Mathematics}, number = {2}, pages = {185 -- 232}, publisher = {Springer}, title = {{Hot topics in cold gases: A mathematical physics perspective}}, doi = {10.1007/s11537-013-1264-5}, volume = {8}, year = {2013}, } @article{2300, abstract = {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.}, author = {Giuliani, Alessandro and Lieb, Élliott and Seiringer, Robert}, journal = {Physical Review B}, number = {6}, publisher = {American Physical Society}, title = {{Realization of stripes and slabs in two and three dimensions}}, doi = {10.1103/PhysRevB.88.064401}, volume = {88}, year = {2013}, } @article{2318, abstract = {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. }, author = {Seiringer, Robert}, journal = {Journal of Spectral Theory}, number = {3}, pages = {321--328}, publisher = {European Mathematical Society}, title = {{Absence of bound states implies non-negativity of the scattering length}}, doi = {10.4171/JST/31}, volume = {2}, year = {2012}, }