TY - JOUR AB - We study the norm approximation to the Schrödinger dynamics of N bosons in with an interaction potential of the form . Assuming that in the initial state the particles outside of the condensate form a quasi-free state with finite kinetic energy, we show that in the large N limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation for all . The range of β is expected to be optimal for this large class of initial states. AU - Nam, Phan AU - Napiórkowski, Marcin M ID - 739 IS - 5 JF - Journal de Mathématiques Pures et Appliquées SN - 00217824 TI - A note on the validity of Bogoliubov correction to mean field dynamics VL - 108 ER - TY - JOUR AB - Recently it was shown that molecules rotating in superfluid helium can be described in terms of the angulon quasiparticles (Phys. Rev. Lett. 118, 095301 (2017)). Here we demonstrate that in the experimentally realized regime the angulon can be seen as a point charge on a 2-sphere interacting with a gauge field of a non-abelian magnetic monopole. Unlike in several other settings, the gauge fields of the angulon problem emerge in the real coordinate space, as opposed to the momentum space or some effective parameter space. Furthermore, we find a topological transition associated with making the monopole abelian, which takes place in the vicinity of the previously reported angulon instabilities. These results pave the way for studying topological phenomena in experiments on molecules trapped in superfluid helium nanodroplets, as well as on other realizations of orbital impurity problems. AU - Yakaboylu, Enderalp AU - Deuchert, Andreas AU - Lemeshko, Mikhail ID - 997 IS - 23 JF - Physical Review Letters SN - 0031-9007 TI - Emergence of non-abelian magnetic monopoles in a quantum impurity problem VL - 119 ER - TY - JOUR AB - We consider a many-body system of fermionic atoms interacting via a local pair potential and subject to an external potential within the framework of Bardeen-Cooper-Schrieffer (BCS) theory. We measure the free energy of the whole sample with respect to the free energy of a reference state which allows us to define a BCS functional with boundary conditions at infinity. Our main result is a lower bound for this energy functional in terms of expressions that typically appear in Ginzburg-Landau functionals. AU - Deuchert, Andreas ID - 912 IS - 8 JF - Journal of Mathematical Physics SN - 00222488 TI - A lower bound for the BCS functional with boundary conditions at infinity VL - 58 ER - TY - JOUR AB - We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive nonlinear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson\'s lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present. AU - Nam, Phan AU - Rougerie, Nicolas AU - Seiringer, Robert ID - 1143 IS - 2 JF - Analysis and PDE TI - Ground states of large bosonic systems: The gross Pitaevskii limit revisited VL - 9 ER - TY - JOUR AB - 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. AU - Bräunlich, Gerhard AU - Hainzl, Christian AU - Seiringer, Robert ID - 1259 IS - 2 JF - Mathematical Physics, Analysis and Geometry TI - Bogolubov–Hartree–Fock theory for strongly interacting fermions in the low density limit VL - 19 ER - TY - JOUR AB - We give a simplified proof of the nonexistence of large nuclei in the liquid drop model and provide an explicit bound. Our bound is within a factor of 2.3 of the conjectured value and seems to be the first quantitative result. AU - Frank, Rupert AU - Killip, Rowan AU - Nam, Phan ID - 1267 IS - 8 JF - Letters in Mathematical Physics TI - Nonexistence of large nuclei in the liquid drop model VL - 106 ER - TY - JOUR AB - We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value Jc(p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of Jc(p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity. AU - Giuliani, Alessandro AU - Seiringer, Robert ID - 1291 IS - 3 JF - Communications in Mathematical Physics TI - Periodic striped ground states in Ising models with competing interactions VL - 347 ER - TY - CONF AB - We report on a mathematically rigorous analysis of the superfluid properties of a Bose- Einstein condensate in the many-body ground state of a one-dimensional model of interacting bosons in a random potential. AU - Könenberg, Martin AU - Moser, Thomas AU - Seiringer, Robert AU - Yngvason, Jakob ID - 1428 IS - 1 T2 - Journal of Physics: Conference Series TI - Superfluidity and BEC in a Model of Interacting Bosons in a Random Potential VL - 691 ER - TY - JOUR AB - We study the time-dependent Bogoliubov–de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg–Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time. The full non-linear structure of the equations is necessary to understand this behavior. AU - Frank, Rupert AU - Hainzl, Christian AU - Schlein, Benjamin AU - Seiringer, Robert ID - 1422 IS - 7 JF - Letters in Mathematical Physics TI - Incompatibility of time-dependent Bogoliubov–de-Gennes and Ginzburg–Landau equations VL - 106 ER - TY - JOUR AB - We study the time evolution of a system of N spinless fermions in R3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system. AU - Bach, Volker AU - Breteaux, Sébastien AU - Petrat, Sören P AU - Pickl, Peter AU - Tzaneteas, Tim ID - 1436 IS - 1 JF - Journal de Mathématiques Pures et Appliquées TI - Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction VL - 105 ER - TY - JOUR AB - We consider the Tonks-Girardeau gas subject to a random external potential. If the disorder is such that the underlying one-particle Hamiltonian displays localization (which is known to be generically the case), we show that there is exponential decay of correlations in the many-body eigenstates. Moreover, there is no Bose-Einstein condensation and no superfluidity, even at zero temperature. AU - Seiringer, Robert AU - Warzel, Simone ID - 1478 IS - 3 JF - New Journal of Physics TI - Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas VL - 18 ER - TY - JOUR AB - We review recent results concerning the mathematical properties of the Bardeen-Cooper-Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers, partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J. P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg-Landau model can be derived from the BCS theory in a suitable parameter regime. AU - Hainzl, Christian AU - Seiringer, Robert ID - 1486 IS - 2 JF - Journal of Mathematical Physics TI - The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties VL - 57 ER - TY - JOUR AB - We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151–164 2011) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence. AU - Petrat, Sören P AU - Pickl, Peter ID - 1493 IS - 1 JF - Mathematical Physics, Analysis and Geometry TI - A new method and a new scaling for deriving fermionic mean-field dynamics VL - 19 ER - TY - JOUR AB - We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive non-linear Schrödinger ground state. AU - Lewin, Mathieu AU - Nam, Phan AU - Rougerie, Nicolas ID - 1491 IS - 9 JF - Transactions of the American Mathematical Society TI - The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases VL - 368 ER - TY - JOUR AB - We provide general conditions for which bosonic quadratic Hamiltonians on Fock spaces can be diagonalized by Bogoliubov transformations. Our results cover the case when quantum systems have infinite degrees of freedom and the associated one-body kinetic and paring operators are unbounded. Our sufficient conditions are optimal in the sense that they become necessary when the relevant one-body operators commute. AU - Nam, Phan AU - Napiórkowski, Marcin M AU - Solovej, Jan ID - 1545 IS - 11 JF - Journal of Functional Analysis TI - Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations VL - 270 ER - TY - JOUR AB - We consider the Bardeen–Cooper–Schrieffer free energy functional for particles interacting via a two-body potential on a microscopic scale and in the presence of weak external fields varying on a macroscopic scale. We study the influence of the external fields on the critical temperature. We show that in the limit where the ratio between the microscopic and macroscopic scale tends to zero, the next to leading order of the critical temperature is determined by the lowest eigenvalue of the linearization of the Ginzburg–Landau equation. AU - Frank, Rupert AU - Hainzl, Christian AU - Seiringer, Robert AU - Solovej, Jan ID - 1620 IS - 1 JF - Communications in Mathematical Physics TI - The external field dependence of the BCS critical temperature VL - 342 ER - TY - JOUR AB - We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases. AU - Lundholm, Douglas AU - Nam, Phan AU - Portmann, Fabian ID - 1622 IS - 3 JF - Archive for Rational Mechanics and Analysis TI - Fractional Hardy–Lieb–Thirring and related Inequalities for interacting systems VL - 219 ER - TY - JOUR AB - 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. AU - Correggi, Michele AU - Giuliani, Alessandro AU - Seiringer, Robert ID - 1572 IS - 1 JF - Communications in Mathematical Physics TI - Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet VL - 339 ER - TY - JOUR AB - 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. AU - Chen, Thomas AU - Hainzl, Christian AU - Pavlović, Nataša AU - Seiringer, Robert ID - 1573 IS - 10 JF - Communications on Pure and Applied Mathematics TI - Unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti VL - 68 ER - TY - JOUR AB - Given a convex function (Formula presented.) and two hermitian matrices A and B, Lewin and Sabin study in (Lett Math Phys 104:691–705, 2014) the relative entropy defined by (Formula presented.). Among other things, they prove that the so-defined quantity is monotone if and only if (Formula presented.) is operator monotone. The monotonicity is then used to properly define (Formula presented.) for bounded self-adjoint operators acting on an infinite-dimensional Hilbert space by a limiting procedure. More precisely, for an increasing sequence of finite-dimensional projections (Formula presented.) with (Formula presented.) strongly, the limit (Formula presented.) is shown to exist and to be independent of the sequence of projections (Formula presented.). The question whether this sequence converges to its "obvious" limit, namely (Formula presented.), has been left open. We answer this question in principle affirmatively and show that (Formula presented.). If the operators A and B are regular enough, that is (A − B), (Formula presented.) and (Formula presented.) are trace-class, the identity (Formula presented.) holds. AU - Deuchert, Andreas AU - Hainzl, Christian AU - Seiringer, Robert ID - 1704 IS - 10 JF - Letters in Mathematical Physics TI - Note on a family of monotone quantum relative entropies VL - 105 ER - TY - JOUR AB - We study a double Cahn-Hilliard type functional related to the Gross-Pitaevskii energy of two-components Bose-Einstein condensates. In the case of large but same order intercomponent and intracomponent coupling strengths, we prove Γ-convergence to a perimeter minimisation functional with an inhomogeneous surface tension. We study the asymptotic behavior of the surface tension as the ratio between the intercomponent and intracomponent coupling strengths becomes very small or very large and obtain good agreement with the physical literature. We obtain as a consequence, symmetry breaking of the minimisers for the harmonic potential. AU - Goldman, Michael AU - Royo-Letelier, Jimena ID - 1807 IS - 3 JF - ESAIM - Control, Optimisation and Calculus of Variations TI - Sharp interface limit for two components Bose-Einstein condensates VL - 21 ER - TY - JOUR AB - We investigate the relation between Bose-Einstein condensation (BEC) and superfluidity in the ground state of a one-dimensional model of interacting bosons in a strong random potential. We prove rigorously that in a certain parameter regime the superfluid fraction can be arbitrarily small while complete BEC prevails. In another regime there is both complete BEC and complete superfluidity, despite the strong disorder AU - Könenberg, Martin AU - Moser, Thomas AU - Seiringer, Robert AU - Yngvason, Jakob ID - 1880 JF - New Journal of Physics TI - Superfluid behavior of a Bose-Einstein condensate in a random potential VL - 17 ER - TY - JOUR AB - We study the spectrum of a large system of N identical bosons interacting via a two-body potential with strength 1/N. In this mean-field regime, Bogoliubov's theory predicts that the spectrum of the N-particle Hamiltonian can be approximated by that of an effective quadratic Hamiltonian acting on Fock space, which describes the fluctuations around a condensed state. Recently, Bogoliubov's theory has been justified rigorously in the case that the low-energy eigenvectors of the N-particle Hamiltonian display complete condensation in the unique minimizer of the corresponding Hartree functional. In this paper, we shall justify Bogoliubov's theory for the high-energy part of the spectrum of the N-particle Hamiltonian corresponding to (non-linear) excited states of the Hartree functional. Moreover, we shall extend the existing results on the excitation spectrum to the case of non-uniqueness and/or degeneracy of the Hartree minimizer. In particular, the latter covers the case of rotating Bose gases, when the rotation speed is large enough to break the symmetry and to produce multiple quantized vortices in the Hartree minimizer. AU - Nam, Phan AU - Seiringer, Robert ID - 2085 IS - 2 JF - Archive for Rational Mechanics and Analysis TI - Collective excitations of Bose gases in the mean-field regime VL - 215 ER - TY - JOUR AB - 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. AU - Lewin, Mathieu AU - Phan Thanh, Nam AU - Rougerie, Nicolas ID - 473 JF - Journal de l'Ecole Polytechnique - Mathematiques TI - Derivation of nonlinear gibbs measures from many-body quantum mechanics VL - 2 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 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 JF - Communications in Mathematical Physics SN - 0010-3616 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 - JOUR AB - 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. AU - Seiringer, Robert ID - 10814 JF - Jahresbericht der Deutschen Mathematiker-Vereinigung KW - General Medicine SN - 0012-0456 TI - The excitation spectrum for Bose fluids with weak interactions VL - 116 ER - TY - CONF AB - 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. AU - Seiringer, Robert ID - 8044 SN - 9788961058063 T2 - Proceeding of the International Congress of Mathematicans TI - Structure of the excitation spectrum for many-body quantum systems VL - 3 ER - TY - JOUR AB - 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. AU - Guo, Yujin AU - Seiringer, Robert ID - 2281 IS - 2 JF - Letters in Mathematical Physics TI - On the mass concentration for Bose-Einstein condensates with attractive interactions VL - 104 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 -