@article{7100, abstract = {We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting froman interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx), for any β>0, or to be given by VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.}, author = {Jeblick, Maximilian and Leopold, Nikolai K and Pickl, Peter}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, number = {1}, pages = {1--69}, publisher = {Springer Nature}, title = {{Derivation of the time dependent Gross–Pitaevskii equation in two dimensions}}, doi = {10.1007/s00220-019-03599-x}, volume = {372}, year = {2019}, } @article{7226, author = {Jaksic, Vojkan and Seiringer, Robert}, issn = {0022-2488}, journal = {Journal of Mathematical Physics}, number = {12}, publisher = {AIP Publishing}, title = {{Introduction to the Special Collection: International Congress on Mathematical Physics (ICMP) 2018}}, doi = {10.1063/1.5138135}, volume = {60}, year = {2019}, } @article{7413, abstract = {We consider Bose gases consisting of N particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of order N−1 (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as N→∞. Our results confirm Bogoliubov’s predictions.}, author = {Boccato, Chiara and Brennecke, Christian and Cenatiempo, Serena and Schlein, Benjamin}, issn = {1871-2509}, journal = {Acta Mathematica}, number = {2}, pages = {219--335}, publisher = {International Press of Boston}, title = {{Bogoliubov theory in the Gross–Pitaevskii limit}}, doi = {10.4310/acta.2019.v222.n2.a1}, volume = {222}, year = {2019}, } @unpublished{7524, abstract = {We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\rho$ and inverse temperature $\beta$ differs from the one of the non-interacting system by the correction term $4 \pi \rho^2 |\ln a^2 \rho|^{-1} (2 - [1 - \beta_{\mathrm{c}}/\beta]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\cdot]_+ = \max\{ 0, \cdot \}$ and $\beta_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit $a^2\rho \ll 1$ and if $\beta \rho \gtrsim 1$.}, author = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert}, booktitle = {arXiv}, pages = {61}, title = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}}, doi = {10.48550/arXiv.1910.03372}, year = {2019}, } @article{5856, abstract = {We give a bound on the ground-state energy of a system of N non-interacting fermions in a three-dimensional cubic box interacting with an impurity particle via point interactions. We show that the change in energy compared to the system in the absence of the impurity is bounded in terms of the gas density and the scattering length of the interaction, independently of N. Our bound holds as long as the ratio of the mass of the impurity to the one of the gas particles is larger than a critical value m∗ ∗≈ 0.36 , which is the same regime for which we recently showed stability of the system.}, author = {Moser, Thomas and Seiringer, Robert}, issn = {1424-0637}, journal = {Annales Henri Poincare}, number = {4}, pages = {1325–1365}, publisher = {Springer}, title = {{Energy contribution of a point-interacting impurity in a Fermi gas}}, doi = {10.1007/s00023-018-00757-0}, volume = {20}, year = {2019}, } @article{295, abstract = {We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter (Formula presented.). The lower bounds extend to Lieb–Thirring inequalities for all anyons except bosons.}, author = {Lundholm, Douglas and Seiringer, Robert}, journal = {Letters in Mathematical Physics}, number = {11}, pages = {2523--2541}, publisher = {Springer}, title = {{Fermionic behavior of ideal anyons}}, doi = {10.1007/s11005-018-1091-y}, volume = {108}, year = {2018}, } @article{180, abstract = {In this paper we define and study the classical Uniform Electron Gas (UEG), a system of infinitely many electrons whose density is constant everywhere in space. The UEG is defined differently from Jellium, which has a positive constant background but no constraint on the density. We prove that the UEG arises in Density Functional Theory in the limit of a slowly varying density, minimizing the indirect Coulomb energy. We also construct the quantum UEG and compare it to the classical UEG at low density.}, author = {Lewi, Mathieu and Lieb, Élliott and Seiringer, Robert}, issn = {2270-518X}, journal = {Journal de l'Ecole Polytechnique - Mathematiques}, pages = {79 -- 116}, publisher = {Ecole Polytechnique}, title = {{Statistical mechanics of the uniform electron gas}}, doi = {10.5802/jep.64}, volume = {5}, year = {2018}, } @inproceedings{11, abstract = {We report on a novel strategy to derive mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. The technique combines the method of counting and the coherent state approach to study the growth of the correlations among the particles and in the radiation field. As an instructional example, we derive the Schrödinger–Klein–Gordon system of equations from the Nelson model with ultraviolet cutoff and possibly massless scalar field. In particular, we prove the convergence of the reduced density matrices (of the nonrelativistic particles and the field bosons) associated with the exact time evolution to the projectors onto the solutions of the Schrödinger–Klein–Gordon equations in trace norm. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the nonrelativistic particles in Sobolev norm.}, author = {Leopold, Nikolai K and Pickl, Peter}, location = {Munich, Germany}, pages = {185 -- 214}, publisher = {Springer}, title = {{Mean-field limits of particles in interaction with quantised radiation fields}}, doi = {10.1007/978-3-030-01602-9_9}, volume = {270}, year = {2018}, } @article{554, abstract = {We analyse the canonical Bogoliubov free energy functional in three dimensions at low temperatures in the dilute limit. We prove existence of a first-order phase transition and, in the limit (Formula presented.), we determine the critical temperature to be (Formula presented.) to leading order. Here, (Formula presented.) is the critical temperature of the free Bose gas, ρ is the density of the gas and a is the scattering length of the pair-interaction potential V. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee–Huang–Yang formula in the limit (Formula presented.).}, author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan}, issn = {0010-3616}, journal = {Communications in Mathematical Physics}, number = {1}, pages = {347--403}, publisher = {Springer}, title = {{The Bogoliubov free energy functional II: The dilute Limit}}, doi = {10.1007/s00220-017-3064-x}, volume = {360}, year = {2018}, } @article{5983, abstract = {We study a quantum impurity possessing both translational and internal rotational degrees of freedom interacting with a bosonic bath. Such a system corresponds to a “rotating polaron,” which can be used to model, e.g., a rotating molecule immersed in an ultracold Bose gas or superfluid helium. We derive the Hamiltonian of the rotating polaron and study its spectrum in the weak- and strong-coupling regimes using a combination of variational, diagrammatic, and mean-field approaches. We reveal how the coupling between linear and angular momenta affects stable quasiparticle states, and demonstrate that internal rotation leads to an enhanced self-localization in the translational degrees of freedom.}, author = {Yakaboylu, Enderalp and Midya, Bikashkali and Deuchert, Andreas and Leopold, Nikolai K and Lemeshko, Mikhail}, issn = {2469-9969}, journal = {Physical Review B}, number = {22}, publisher = {American Physical Society}, title = {{Theory of the rotating polaron: Spectrum and self-localization}}, doi = {10.1103/physrevb.98.224506}, volume = {98}, year = {2018}, } @article{6002, abstract = {The Bogoliubov free energy functional is analysed. The functional serves as a model of a translation-invariant Bose gas at positive temperature. We prove the existence of minimizers in the case of repulsive interactions given by a sufficiently regular two-body potential. Furthermore, we prove the existence of a phase transition in this model and provide its phase diagram.}, author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan Philip}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {3}, pages = {1037--1090}, publisher = {Springer Nature}, title = {{The Bogoliubov free energy functional I: Existence of minimizers and phase diagram}}, doi = {10.1007/s00205-018-1232-6}, volume = {229}, year = {2018}, } @article{399, abstract = {Following an earlier calculation in 3D, we calculate the 2D critical temperature of a dilute, translation-invariant Bose gas using a variational formulation of the Bogoliubov approximation introduced by Critchley and Solomon in 1976. This provides the first analytical calculation of the Kosterlitz-Thouless transition temperature that includes the constant in the logarithm.}, author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan}, journal = {EPL}, number = {1}, publisher = {IOP Publishing}, title = {{Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation}}, doi = {10.1209/0295-5075/121/10007}, volume = {121}, year = {2018}, } @article{400, abstract = {We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. In the case of vanishing angular momentum, our results carry over to the three-dimensional case.}, author = {Deuchert, Andreas and Geisinge, Alissa and Hainzl, Christian and Loss, Michael}, journal = {Annales Henri Poincare}, number = {5}, pages = {1507 -- 1527}, publisher = {Springer}, title = {{Persistence of translational symmetry in the BCS model with radial pair interaction}}, doi = {10.1007/s00023-018-0665-7}, volume = {19}, year = {2018}, } @article{446, abstract = {We prove that in Thomas–Fermi–Dirac–von Weizsäcker theory, a nucleus of charge Z > 0 can bind at most Z + C electrons, where C is a universal constant. This result is obtained through a comparison with Thomas-Fermi theory which, as a by-product, gives bounds on the screened nuclear potential and the radius of the minimizer. A key ingredient of the proof is a novel technique to control the particles in the exterior region, which also applies to the liquid drop model with a nuclear background potential.}, author = {Frank, Rupert and Phan Thanh, Nam and Van Den Bosch, Hanne}, journal = {Communications on Pure and Applied Mathematics}, number = {3}, pages = {577 -- 614}, publisher = {Wiley-Blackwell}, title = {{The ionization conjecture in Thomas–Fermi–Dirac–von Weizsäcker theory}}, doi = {10.1002/cpa.21717}, volume = {71}, year = {2018}, } @article{455, abstract = {The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose–Einstein condensation and quantum chemistry. We reformulate the Dirac–Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov–de Gennes and Hartree–Fock–Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov–de Gennes equations in energy space and discuss conserved quantities}, author = {Benedikter, Niels P and Sok, Jérémy and Solovej, Jan}, journal = {Annales Henri Poincare}, number = {4}, pages = {1167 -- 1214}, publisher = {Birkhäuser}, title = {{The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations}}, doi = {10.1007/s00023-018-0644-z}, volume = {19}, year = {2018}, } @phdthesis{52, abstract = {In this thesis we will discuss systems of point interacting fermions, their stability and other spectral properties. Whereas for bosons a point interacting system is always unstable this ques- tion is more subtle for a gas of two species of fermions. In particular the answer depends on the mass ratio between these two species. Most of this work will be focused on the N + M model which consists of two species of fermions with N, M particles respectively which interact via point interactions. We will introduce this model using a formal limit and discuss the N + 1 system in more detail. In particular, we will show that for mass ratios above a critical one, which does not depend on the particle number, the N + 1 system is stable. In the context of this model we will prove rigorous versions of Tan relations which relate various quantities of the point-interacting model. By restricting the N + 1 system to a box we define a finite density model with point in- teractions. In the context of this system we will discuss the energy change when introducing a point-interacting impurity into a system of non-interacting fermions. We will see that this change in energy is bounded independently of the particle number and in particular the bound only depends on the density and the scattering length. As another special case of the N + M model we will show stability of the 2 + 2 model for mass ratios in an interval around one. Further we will investigate a different model of point interactions which was discussed before in the literature and which is, contrary to the N + M model, not given by a limiting procedure but is based on a Dirichlet form. We will show that this system behaves trivially in the thermodynamic limit, i.e. the free energy per particle is the same as the one of the non-interacting system.}, author = {Moser, Thomas}, issn = {2663-337X}, pages = {115}, publisher = {Institute of Science and Technology Austria}, title = {{Point interactions in systems of fermions}}, doi = {10.15479/AT:ISTA:th_1043}, year = {2018}, } @article{154, abstract = {We give a lower bound on the ground state energy of a system of two fermions of one species interacting with two fermions of another species via point interactions. We show that there is a critical mass ratio m2 ≈ 0.58 such that the system is stable, i.e., the energy is bounded from below, for m∈[m2,m2−1]. So far it was not known whether this 2 + 2 system exhibits a stable region at all or whether the formation of four-body bound states causes an unbounded spectrum for all mass ratios, similar to the Thomas effect. Our result gives further evidence for the stability of the more general N + M system.}, author = {Moser, Thomas and Seiringer, Robert}, issn = {1572-9656}, journal = {Mathematical Physics Analysis and Geometry}, number = {3}, publisher = {Springer}, title = {{Stability of the 2+2 fermionic system with point interactions}}, doi = {10.1007/s11040-018-9275-3}, volume = {21}, year = {2018}, } @article{1079, abstract = {We study the ionization problem in the Thomas-Fermi-Dirac-von Weizsäcker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.}, author = {Nam, Phan and Van Den Bosch, Hanne}, issn = {1385-0172}, journal = {Mathematical Physics, Analysis and Geometry}, number = {2}, publisher = {Springer}, title = {{Nonexistence in Thomas Fermi-Dirac-von Weizsäcker theory with small nuclear charges}}, doi = {10.1007/s11040-017-9238-0}, volume = {20}, year = {2017}, } @article{912, abstract = {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. }, author = {Deuchert, Andreas}, issn = {0022-2488}, journal = { Journal of Mathematical Physics}, number = {8}, publisher = {AIP Publishing}, title = {{A lower bound for the BCS functional with boundary conditions at infinity}}, doi = {10.1063/1.4996580}, volume = {58}, year = {2017}, } @article{484, abstract = {We consider the dynamics of a large quantum system of N identical bosons in 3D interacting via a two-body potential of the form N3β-1w(Nβ(x - y)). For fixed 0 = β < 1/3 and large N, we obtain a norm approximation to the many-body evolution in the Nparticle Hilbert space. The leading order behaviour of the dynamics is determined by Hartree theory while the second order is given by Bogoliubov theory.}, author = {Nam, Phan and Napiórkowski, Marcin M}, issn = {1095-0761}, journal = {Advances in Theoretical and Mathematical Physics}, number = {3}, pages = {683 -- 738}, publisher = {International Press}, title = {{Bogoliubov correction to the mean-field dynamics of interacting bosons}}, doi = {10.4310/ATMP.2017.v21.n3.a4}, volume = {21}, year = {2017}, }