@article{1436,
abstract = {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.},
author = {Bach, Volker and Breteaux, Sébastien and Petrat, Sören P and Pickl, Peter and Tzaneteas, Tim},
journal = {Journal de Mathématiques Pures et Appliquées},
number = {1},
pages = {1 -- 30},
publisher = {Elsevier},
title = {{Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction}},
doi = {10.1016/j.matpur.2015.09.003},
volume = {105},
year = {2016},
}
@article{1478,
abstract = {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.},
author = {Seiringer, Robert and Warzel, Simone},
journal = {New Journal of Physics},
number = {3},
publisher = {IOP Publishing Ltd.},
title = {{Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas}},
doi = {10.1088/1367-2630/18/3/035002},
volume = {18},
year = {2016},
}
@article{1486,
abstract = {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.},
author = {Hainzl, Christian and Seiringer, Robert},
journal = {Journal of Mathematical Physics},
number = {2},
publisher = {American Institute of Physics},
title = {{The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties}},
doi = {10.1063/1.4941723},
volume = {57},
year = {2016},
}
@article{1491,
abstract = {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.},
author = {Lewin, Mathieu and Nam, Phan and Rougerie, Nicolas},
journal = {Transactions of the American Mathematical Society},
number = {9},
pages = {6131 -- 6157},
publisher = {American Mathematical Society},
title = {{The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases}},
doi = {10.1090/tran/6537},
volume = {368},
year = {2016},
}
@article{1493,
abstract = {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.},
author = {Petrat, Sören P and Pickl, Peter},
journal = {Mathematical Physics, Analysis and Geometry},
number = {1},
publisher = {Springer},
title = {{A new method and a new scaling for deriving fermionic mean-field dynamics}},
doi = {10.1007/s11040-016-9204-2},
volume = {19},
year = {2016},
}
@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{1704,
abstract = {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.},
author = {Deuchert, Andreas and Hainzl, Christian and Seiringer, Robert},
journal = {Letters in Mathematical Physics},
number = {10},
pages = {1449 -- 1466},
publisher = {Springer},
title = {{Note on a family of monotone quantum relative entropies}},
doi = {10.1007/s11005-015-0787-5},
volume = {105},
year = {2015},
}
@article{1807,
abstract = {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.},
author = {Goldman, Michael and Royo-Letelier, Jimena},
journal = {ESAIM - Control, Optimisation and Calculus of Variations},
number = {3},
pages = {603 -- 624},
publisher = {EDP Sciences},
title = {{Sharp interface limit for two components Bose-Einstein condensates}},
doi = {10.1051/cocv/2014040},
volume = {21},
year = {2015},
}
@article{1880,
abstract = {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},
author = {Könenberg, Martin and Moser, Thomas and Seiringer, Robert and Yngvason, Jakob},
journal = {New Journal of Physics},
publisher = {IOP Publishing Ltd.},
title = {{Superfluid behavior of a Bose-Einstein condensate in a random potential}},
doi = {10.1088/1367-2630/17/1/013022},
volume = {17},
year = {2015},
}
@article{2085,
abstract = {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. },
author = {Nam, Phan and Seiringer, Robert},
journal = {Archive for Rational Mechanics and Analysis},
number = {2},
pages = {381 -- 417},
publisher = {Springer},
title = {{Collective excitations of Bose gases in the mean-field regime}},
doi = {10.1007/s00205-014-0781-6},
volume = {215},
year = {2015},
}
@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 = {Kyung Moon SA},
title = {{Structure of the excitation spectrum for many-body quantum systems}},
volume = {3},
year = {2014},
}
@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 = {0012-0456},
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},
}
@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{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},
}
@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{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{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{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{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},
}