@article{14542, abstract = {It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature Ξ and the critical temperature Tc is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions d=1,2 and give an exemplary proof in the weak coupling limit.}, author = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard and Roos, Barbara}, issn = {1793-6659}, journal = {Reviews in Mathematical Physics}, number = {9}, publisher = {World Scientific Publishing}, title = {{Universality in low-dimensional BCS theory}}, doi = {10.1142/s0129055x2360005x}, volume = {36}, year = {2024}, } @article{18107, abstract = {We consider a dilute fully spin-polarized Fermi gas at positive temperature in dimensions d∈{1,2,3} . We show that the pressure of the interacting gas is bounded from below by that of the free gas plus, to leading order, an explicit term of order adρ2+2/d, where a is the p-wave scattering length of the repulsive interaction and ρ is the particle density. The results are valid for a wide range of repulsive interactions, including that of a hard core, and uniform in temperatures at most of the order of the Fermi temperature. A central ingredient in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237–260).}, author = {Lauritsen, Asbjørn Bækgaard and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Pressure of a dilute spin-polarized Fermi gas: Lower bound}}, doi = {10.1017/fms.2024.56}, volume = {12}, year = {2024}, } @phdthesis{18135, abstract = {This thesis consists of two separate parts. In the first part we consider a dilute Fermi gas interacting through a repulsive interaction in dimensions $d=1,2,3$. Our focus is mostly on the physically most relevant dimension $d=3$ and the setting of a spin-polarized (equivalently spinless) gas, where the Pauli exclusion principle plays a key role. We show that, at zero temperature, the ground state energy density of the interacting spin-polarized gas differs (to leading order) from that of the free (i.e. non-interacting) gas by a term of order $a_p^d\rho^{2+2/d}$ with $a_p$ the $p$-wave scattering length of the repulsive interaction and $\rho$ the density. Further, we extend this to positive temperature and show that the pressure of an interacting spin-polarized gas differs from that of the free gas by a now temperature dependent term, again of order $a_p^d\rho^{2+2/d}$. Lastly, we consider the setting of a spin-$\frac{1}{2}$ Fermi gas in $d=3$ dimensions and show that here, as an upper bound, the ground state energy density differs from that of the free system by a term of order $a_s \rho^2$ with an error smaller than $a_s \rho^2 (a_s\rho^{1/3})^{1-\eps}$ for any $\eps > 0$, where $a_s$ is the $s$-wave scattering length of the repulsive interaction. These asymptotic formulas complement the similar formulas in the literature for the dilute Bose and spin-$\frac{1}{2}$ Fermi gas, where the ground state energies or pressures differ from that of the corresponding free systems by a term of order $a_s \rho^2$ in dimension $d=3$. In the spin-polarized setting, the corrections, of order $a_p^3\rho^{8/3}$ in dimension $d=3$, are thus much smaller and requires a more delicate analysis. In the second part of the thesis we consider the Bardeen--Cooper--Schrieffer (BCS) theory of superconductivity and in particular its associated critical temperature and energy gap. We prove that the ratio of the zero-temperature energy gap and critical temperature $\Xi(T=0)/T_c$ approaches a universal constant $\pi e^{-\gamma}\approx 1.76$ in both the limit of high density in dimension $d=3$ and in the limit of weak coupling in dimensions $d=1,2$. This complements the proofs in the literature of this universal behaviour in the limit of weak coupling or low density in dimension $d=3$. Secondly, we prove that the ratio of the energy gap at positive temperature and critical temperature $\Xi(T)/T_c$ approaches a universal function of the relative temperature $T/T_c$ in the limit of weak coupling in dimensions $d=1,2,3$.}, author = {Lauritsen, Asbjørn Bækgaard}, isbn = {978-3-99078-042-8}, issn = {2663-337X}, pages = {353}, publisher = {Institute of Science and Technology Austria}, title = {{Energies of dilute Fermi gases and universalities in BCS theory}}, doi = {10.15479/at:ista:18135}, year = {2024}, } @article{14931, abstract = {We prove an upper bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions. One of the main ingredients in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin et al. (1971) [15].}, author = {Lauritsen, Asbjørn Bækgaard and Seiringer, Robert}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {7}, publisher = {Elsevier}, title = {{Ground state energy of the dilute spin-polarized Fermi gas: Upper bound via cluster expansion}}, doi = {10.1016/j.jfa.2024.110320}, volume = {286}, year = {2024}, } @article{12183, abstract = {We consider a gas of n bosonic particles confined in a box [−ℓ/2,ℓ/2]3 with Neumann boundary conditions. We prove Bose–Einstein condensation in the Gross–Pitaevskii regime, with an optimal bound on the condensate depletion. Moreover, our lower bound for the ground state energy in a small box [−ℓ/2,ℓ/2]3 implies (via Neumann bracketing) a lower bound for the ground state energy of N bosons in a large box [−L/2,L/2]3 with density ρ=N/L3 in the thermodynamic limit.}, author = {Boccato, Chiara and Seiringer, Robert}, issn = {1424-0637}, journal = {Annales Henri Poincare}, pages = {1505--1560}, publisher = {Springer Nature}, title = {{The Bose Gas in a box with Neumann boundary conditions}}, doi = {10.1007/s00023-022-01252-3}, volume = {24}, year = {2023}, } @article{14192, abstract = {For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the translation invariant Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling.}, author = {Lampart, Jonas and Mitrouskas, David Johannes and Mysliwy, Krzysztof}, issn = {1572-9656}, journal = {Mathematical Physics, Analysis and Geometry}, keywords = {Geometry and Topology, Mathematical Physics}, number = {3}, publisher = {Springer Nature}, title = {{On the global minimum of the energy–momentum relation for the polaron}}, doi = {10.1007/s11040-023-09460-x}, volume = {26}, year = {2023}, } @article{14254, abstract = {In [10] Nam proved a Lieb–Thirring Inequality for the kinetic energy of a fermionic quantum system, with almost optimal (semi-classical) constant and a gradient correction term. We present a stronger version of this inequality, with a much simplified proof. As a corollary we obtain a simple proof of the original Lieb–Thirring inequality.}, author = {Seiringer, Robert and Solovej, Jan Philip}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {10}, publisher = {Elsevier}, title = {{A simple approach to Lieb-Thirring type inequalities}}, doi = {10.1016/j.jfa.2023.110129}, volume = {285}, year = {2023}, } @article{14441, abstract = {We study the Fröhlich polaron model in R3, and establish the subleading term in the strong coupling asymptotics of its ground state energy, corresponding to the quantum corrections to the classical energy determined by the Pekar approximation.}, author = {Brooks, Morris and Seiringer, Robert}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {287--337}, publisher = {Springer Nature}, title = {{The Fröhlich Polaron at strong coupling: Part I - The quantum correction to the classical energy}}, doi = {10.1007/s00220-023-04841-3}, volume = {404}, year = {2023}, } @article{14662, abstract = {We consider a class of polaron models, including the Fröhlich model, at zero total momentum, and show that at sufficiently weak coupling there are no excited eigenvalues below the essential spectrum.}, author = {Seiringer, Robert}, issn = {1664-0403}, journal = {Journal of Spectral Theory}, number = {3}, pages = {1045--1055}, publisher = {EMS Press}, title = {{Absence of excited eigenvalues for Fröhlich type polaron models at weak coupling}}, doi = {10.4171/JST/469}, volume = {13}, year = {2023}, } @article{14715, abstract = {We consider N trapped bosons in the mean-field limit with coupling constant λN = 1/(N − 1). The ground state of such systems exhibits Bose–Einstein condensation. We prove that the probability of finding ℓ particles outside the condensate wave function decays exponentially in ℓ.}, author = {Mitrouskas, David Johannes and Pickl, Peter}, issn = {1089-7658}, journal = {Journal of Mathematical Physics}, number = {12}, publisher = {AIP Publishing}, title = {{Exponential decay of the number of excitations in the weakly interacting Bose gas}}, doi = {10.1063/5.0172199}, volume = {64}, year = {2023}, } @article{12430, abstract = {We study the time evolution of the Nelson model in a mean-field limit in which N nonrelativistic bosons weakly couple (with respect to the particle number) to a positive or zero mass quantized scalar field. Our main result is the derivation of the Bogoliubov dynamics and higher-order corrections. More precisely, we prove the convergence of the approximate wave function to the many-body wave function in norm, with a convergence rate proportional to the number of corrections taken into account in the approximation. We prove an analogous result for the unitary propagator. As an application, we derive a simple system of partial differential equations describing the time evolution of the first- and second-order approximations to the one-particle reduced density matrices of the particles and the quantum field, respectively.}, author = {Falconi, Marco and Leopold, Nikolai K and Mitrouskas, David Johannes and Petrat, Sören P}, issn = {0129-055X}, journal = {Reviews in Mathematical Physics}, number = {4}, publisher = {World Scientific Publishing}, title = {{Bogoliubov dynamics and higher-order corrections for the regularized Nelson model}}, doi = {10.1142/S0129055X2350006X}, volume = {35}, year = {2023}, } @misc{12869, abstract = {We introduce a stochastic cellular automaton as a model for culture and border formation. The model can be conceptualized as a game where the expansion rate of cultures is quantified in terms of their area and perimeter in such a way that approximately round cultures get a competitive advantage. We first analyse the model with periodic boundary conditions, where we study how the model can end up in a fixed state, i.e. freezes. Then we implement the model on the European geography with mountains and rivers. We see how the model reproduces some qualitative features of European culture formation, namely that rivers and mountains are more frequently borders between cultures, mountainous regions tend to have higher cultural diversity and the central European plain has less clear cultural borders. }, author = {Klausen, Frederik Ravn and Lauritsen, Asbjørn Bækgaard}, publisher = {Institute of Science and Technology Austria}, title = {{Research data for: A stochastic cellular automaton model of culture formation}}, doi = {10.15479/AT:ISTA:12869}, year = {2023}, } @article{12890, abstract = {We introduce a stochastic cellular automaton as a model for culture and border formation. The model can be conceptualized as a game where the expansion rate of cultures is quantified in terms of their area and perimeter in such a way that approximately geometrically round cultures get a competitive advantage. We first analyze the model with periodic boundary conditions, where we study how the model can end up in a fixed state, i.e., freezes. Then we implement the model on the European geography with mountains and rivers. We see how the model reproduces some qualitative features of European culture formation, namely, that rivers and mountains are more frequently borders between cultures, mountainous regions tend to have higher cultural diversity, and the central European plain has less clear cultural borders.}, author = {Klausen, Frederik Ravn and Lauritsen, Asbjørn Bækgaard}, issn = {2470-0053}, journal = {Physical Review E}, number = {5}, publisher = {American Physical Society}, title = {{Stochastic cellular automaton model of culture formation}}, doi = {10.1103/PhysRevE.108.054307}, volume = {108}, year = {2023}, } @article{12911, abstract = {This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite-dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground-state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.}, author = {Feliciangeli, Dario and Gerolin, Augusto and Portinale, Lorenzo}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {4}, publisher = {Elsevier}, title = {{A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature}}, doi = {10.1016/j.jfa.2023.109963}, volume = {285}, year = {2023}, } @article{13178, abstract = {We consider the large polaron described by the Fröhlich Hamiltonian and study its energy-momentum relation defined as the lowest possible energy as a function of the total momentum. Using a suitable family of trial states, we derive an optimal parabolic upper bound for the energy-momentum relation in the limit of strong coupling. The upper bound consists of a momentum independent term that agrees with the predicted two-term expansion for the ground state energy of the strongly coupled polaron at rest and a term that is quadratic in the momentum with coefficient given by the inverse of twice the classical effective mass introduced by Landau and Pekar.}, author = {Mitrouskas, David Johannes and Mysliwy, Krzysztof and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics}, pages = {1--52}, publisher = {Cambridge University Press}, title = {{Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron}}, doi = {10.1017/fms.2023.45}, volume = {11}, year = {2023}, } @article{13225, abstract = {Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only |⋅|V^∈ℓ1(Z3). Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy.}, author = {Benedikter, Niels P and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {4}, publisher = {Springer Nature}, title = {{Correlation energy of a weakly interacting Fermi gas with large interaction potential}}, doi = {10.1007/s00205-023-01893-6}, volume = {247}, year = {2023}, } @article{13226, abstract = {We consider the ground state and the low-energy excited states of a system of N identical bosons with interactions in the mean-field scaling regime. For the ground state, we derive a weak Edgeworth expansion for the fluctuations of bounded one-body operators, which yields corrections to a central limit theorem to any order in 1/N−−√. For suitable excited states, we show that the limiting distribution is a polynomial times a normal distribution, and that higher-order corrections are given by an Edgeworth-type expansion.}, author = {Bossmann, Lea and Petrat, Sören P}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, number = {4}, publisher = {Springer Nature}, title = {{Weak Edgeworth expansion for the mean-field Bose gas}}, doi = {10.1007/s11005-023-01698-4}, volume = {113}, year = {2023}, } @article{14854, abstract = {We study the spectrum of the Fröhlich Hamiltonian for the polaron at fixed total momentum. We prove the existence of excited eigenvalues between the ground state energy and the essential spectrum at strong coupling. In fact, our main result shows that the number of excited energy bands diverges in the strong coupling limit. To prove this we derive upper bounds for the min-max values of the corresponding fiber Hamiltonians and compare them with the bottom of the essential spectrum, a lower bound on which was recently obtained by Brooks and Seiringer (Comm. Math. Phys. 404:1 (2023), 287–337). The upper bounds are given in terms of the ground state energy band shifted by momentum-independent excitation energies determined by an effective Hamiltonian of Bogoliubov type.}, author = {Mitrouskas, David Johannes and Seiringer, Robert}, issn = {2578-5885}, journal = {Pure and Applied Analysis}, keywords = {General Medicine}, number = {4}, pages = {973--1008}, publisher = {Mathematical Sciences Publishers}, title = {{Ubiquity of bound states for the strongly coupled polaron}}, doi = {10.2140/paa.2023.5.973}, volume = {5}, year = {2023}, } @inbook{14992, abstract = {In this chapter we first review the Levy–Lieb functional, which gives the lowest kinetic and interaction energy that can be reached with all possible quantum states having a given density. We discuss two possible convex generalizations of this functional, corresponding to using mixed canonical and grand-canonical states, respectively. We present some recent works about the local density approximation, in which the functionals get replaced by purely local functionals constructed using the uniform electron gas energy per unit volume. We then review the known upper and lower bounds on the Levy–Lieb functionals. We start with the kinetic energy alone, then turn to the classical interaction alone, before we are able to put everything together. A later section is devoted to the Hohenberg–Kohn theorem and the role of many-body unique continuation in its proof.}, author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert}, booktitle = {Density Functional Theory}, editor = {Cances, Eric and Friesecke, Gero}, isbn = {9783031223396}, issn = {3005-0286}, pages = {115--182}, publisher = {Springer}, title = {{Universal Functionals in Density Functional Theory}}, doi = {10.1007/978-3-031-22340-2_3}, year = {2023}, } @article{17074, abstract = {We verify Bogoliubov's approximation for translation invariant Bose gases in the mean field regime, i.e. we prove that the ground state energy EN is given by EN=NeH+infσ(H)+oN→∞(1), where N is the number of particles, eH is the minimal Hartree energy and H is the Bogoliubov Hamiltonian. As an intermediate result we show the existence of approximate ground states ΨN, i.e. states satisfying ⟨HN⟩ΨN=EN+oN→∞(1), exhibiting complete Bose--Einstein condensation with respect to one of the Hartree minimizers.}, author = {Brooks, Morris and Seiringer, Robert}, issn = {2690-1005}, journal = {Probability and Mathematical Physics}, number = {4}, pages = {939--1000}, publisher = {Mathematical Sciences Publishers}, title = {{Validity of Bogoliubov’s approximation fortranslation-invariant Bose gases}}, doi = {10.2140/pmp.2022.3.939}, volume = {3}, year = {2023}, }