@article{7790,
  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 𝜌 and inverse temperature 𝛽 differs from the one of the noninteracting system by the correction term 𝜋𝜌𝜌𝛽𝛽 . Here, is the scattering length of the interaction potential, and 𝛽 is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit 𝜌 and if 𝛽𝜌 .},
  author       = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert},
  issn         = {2050-5094},
  journal      = {Forum of Mathematics, Sigma},
  publisher    = {Cambridge University Press},
  title        = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}},
  doi          = {10.1017/fms.2020.17},
  volume       = {8},
  year         = {2020},
}

@article{8705,
  abstract     = {We consider the quantum mechanical many-body problem of a single impurity particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study the Hamiltonian of this system in the mean-field limit and rigorously show that, at low energies, the problem is well described by the Fröhlich polaron model.},
  author       = {Mysliwy, Krzysztof and Seiringer, Robert},
  issn         = {1424-0637},
  journal      = {Annales Henri Poincare},
  number       = {12},
  pages        = {4003--4025},
  publisher    = {Springer Nature},
  title        = {{Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit}},
  doi          = {10.1007/s00023-020-00969-3},
  volume       = {21},
  year         = {2020},
}

@article{9781,
  abstract     = {We consider the Pekar functional on a ball in ℝ3. We prove uniqueness of minimizers, and a quadratic lower bound in terms of the distance to the minimizer. The latter follows from nondegeneracy of the Hessian at the minimum.},
  author       = {Feliciangeli, Dario and Seiringer, Robert},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Applied Mathematics, Computational Mathematics, Analysis},
  number       = {1},
  pages        = {605--622},
  publisher    = {Society for Industrial and Applied Mathematics },
  title        = {{Uniqueness and nondegeneracy of minimizers of the Pekar functional on a ball}},
  doi          = {10.1137/19m126284x},
  volume       = {52},
  year         = {2020},
}

@phdthesis{7514,
  abstract     = {We study the interacting homogeneous Bose gas in two spatial dimensions in the thermodynamic limit at fixed density. We shall be concerned with some mathematical aspects of this complicated problem in many-body quantum mechanics. More specifically, we consider the dilute limit where the scattering length of the interaction potential, which is a measure for the effective range of the potential, is small compared to the average distance between the particles. We are interested in a setting with positive (i.e., non-zero) temperature. After giving a survey of the relevant literature in the field, we provide some facts and examples to set expectations for the two-dimensional system. The crucial difference to the three-dimensional system is that there is no Bose–Einstein condensate at positive temperature due to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic formula for the free energy holds similarly to the three-dimensional case.
We motivate this formula by considering a toy model with δ interaction potential. By restricting this model Hamiltonian to certain trial states with a quasi-condensate we obtain an upper bound for the free energy that still has the quasi-condensate fraction as a free parameter. When minimizing over the quasi-condensate fraction, we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity, which plays an important role in our rigorous contribution. The mathematically rigorous result that we prove concerns the specific free energy in the dilute limit. We give upper and lower bounds on the free energy in terms of the free energy of the non-interacting system and a correction term coming from the interaction. Both bounds match and thus we obtain the leading term of an asymptotic approximation in the dilute limit, provided the thermal wavelength of the particles is of the same order (or larger) than the average distance between the particles. The remarkable feature of this result is its generality: the correction term depends on the interaction potential only through its scattering length and it holds for all nonnegative interaction potentials with finite scattering length that are measurable. In particular, this allows to model an interaction of hard disks.},
  author       = {Mayer, Simon},
  issn         = {2663-337X},
  pages        = {148},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{The free energy of a dilute two-dimensional Bose gas}},
  doi          = {10.15479/AT:ISTA:7514},
  year         = {2020},
}

@article{8587,
  abstract     = {Inspired by the possibility to experimentally manipulate and enhance chemical reactivity in helium nanodroplets, we investigate the effective interaction and the resulting correlations between two diatomic molecules immersed in a bath of bosons. By analogy with the bipolaron, we introduce the biangulon quasiparticle describing two rotating molecules that align with respect to each other due to the effective attractive interaction mediated by the excitations of the bath. We study this system in different parameter regimes and apply several theoretical approaches to describe its properties. Using a Born–Oppenheimer approximation, we investigate the dependence of the effective intermolecular interaction on the rotational state of the two molecules. In the strong-coupling regime, a product-state ansatz shows that the molecules tend to have a strong alignment in the ground state. To investigate the system in the weak-coupling regime, we apply a one-phonon excitation variational ansatz, which allows us to access the energy spectrum. In comparison to the angulon quasiparticle, the biangulon shows shifted angulon instabilities and an additional spectral instability, where resonant angular momentum transfer between the molecules and the bath takes place. These features are proposed as an experimentally observable signature for the formation of the biangulon quasiparticle. Finally, by using products of single angulon and bare impurity wave functions as basis states, we introduce a diagonalization scheme that allows us to describe the transition from two separated angulons to a biangulon as a function of the distance between the two molecules.},
  author       = {Li, Xiang and Yakaboylu, Enderalp and Bighin, Giacomo and Schmidt, Richard and Lemeshko, Mikhail and Deuchert, Andreas},
  issn         = {1089-7690},
  journal      = {The Journal of Chemical Physics},
  keywords     = {Physical and Theoretical Chemistry, General Physics and Astronomy},
  number       = {16},
  publisher    = {AIP Publishing},
  title        = {{Intermolecular forces and correlations mediated by a phonon bath}},
  doi          = {10.1063/1.5144759},
  volume       = {152},
  year         = {2020},
}

@article{6649,
  abstract     = {While Hartree–Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree–Fock state given by plane waves and introduce collective particle–hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann–Brueckner–type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.
},
  author       = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {2097–2150},
  publisher    = {Springer Nature},
  title        = {{Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime}},
  doi          = {10.1007/s00220-019-03505-5},
  volume       = {374},
  year         = {2020},
}

@article{80,
  abstract     = {We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ω, the temperature T, and the particle number N are related by N∼ (T/ ω) 3→ ∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.},
  author       = {Deuchert, Andreas and Seiringer, Robert and Yngvason, Jakob},
  journal      = {Communications in Mathematical Physics},
  number       = {2},
  pages        = {723--776},
  publisher    = {Springer},
  title        = {{Bose–Einstein condensation in a dilute, trapped gas at positive temperature}},
  doi          = {10.1007/s00220-018-3239-0},
  volume       = {368},
  year         = {2019},
}

@article{6788,
  abstract     = {We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant with semiclassical structure and a coherent state, which evolve according to a fermionic version of the Schrödinger–Klein–Gordon equations.},
  author       = {Leopold, Nikolai K and Petrat, Sören P},
  issn         = {1424-0661},
  journal      = {Annales Henri Poincare},
  number       = {10},
  pages        = {3471–3508},
  publisher    = {Springer Nature},
  title        = {{Mean-field dynamics for the Nelson model with fermions}},
  doi          = {10.1007/s00023-019-00828-w},
  volume       = {20},
  year         = {2019},
}

@article{6840,
  abstract     = {We discuss thermodynamic properties of harmonically trapped
imperfect quantum gases. The spatial inhomogeneity of these systems imposes
a redefinition of the mean-field interparticle potential energy as compared
to the homogeneous case. In our approach, it takes the form a
2N2 ωd, where
N is the number of particles, ω—the harmonic trap frequency, d—system’s
dimensionality, and a is a parameter characterizing the interparticle interaction.
We provide arguments that this model corresponds to the limiting case of
a long-ranged interparticle potential of vanishingly small amplitude. This
conclusion is drawn from a computation similar to the well-known Kac scaling
procedure, which is presented here in a form adapted to the case of an isotropic
harmonic trap. We show that within the model, the imperfect gas of trapped
repulsive bosons undergoes the Bose–Einstein condensation provided d > 1.
The main result of our analysis is that in d = 1 the gas of attractive imperfect
fermions with a = −aF < 0 is thermodynamically equivalent to the gas of
repulsive bosons with a = aB > 0 provided the parameters aF and aB fulfill
the relation aB + aF = . This result supplements similar recent conclusion
about thermodynamic equivalence of two-dimensional (2D) uniform imperfect
repulsive Bose and attractive Fermi gases.},
  author       = {Mysliwy, Krzysztof and Napiórkowski, Marek},
  issn         = {1742-5468},
  journal      = {Journal of Statistical Mechanics: Theory and Experiment},
  number       = {6},
  publisher    = {IOP Publishing},
  title        = {{Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps}},
  doi          = {10.1088/1742-5468/ab190d},
  volume       = {2019},
  year         = {2019},
}

@article{7015,
  abstract     = {We modify the "floating crystal" trial state for the classical homogeneous electron gas (also known as jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground-state energy of jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in density functional theory. Finally, in the third system each electron interacts with a periodic image of itself; that is, periodic boundary conditions are imposed on the interaction potential.},
  author       = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert},
  issn         = {2469-9969},
  journal      = {Physical Review B},
  number       = {3},
  publisher    = {American Physical Society},
  title        = {{Floating Wigner crystal with no boundary charge fluctuations}},
  doi          = {10.1103/physrevb.100.035127},
  volume       = {100},
  year         = {2019},
}

@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},
}