TY - JOUR AB - 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]. AU - Lauritsen, Asbjørn Bækgaard AU - Seiringer, Robert ID - 14931 IS - 7 JF - Journal of Functional Analysis SN - 0022-1236 TI - Ground state energy of the dilute spin-polarized Fermi gas: Upper bound via cluster expansion VL - 286 ER - TY - JOUR AB - 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. AU - Boccato, Chiara AU - Seiringer, Robert ID - 12183 JF - Annales Henri Poincare SN - 1424-0637 TI - The Bose Gas in a box with Neumann boundary conditions VL - 24 ER - TY - JOUR AB - 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. AU - Falconi, Marco AU - Leopold, Nikolai K AU - Mitrouskas, David Johannes AU - Petrat, Sören P ID - 12430 IS - 4 JF - Reviews in Mathematical Physics SN - 0129-055X TI - Bogoliubov dynamics and higher-order corrections for the regularized Nelson model VL - 35 ER - TY - THES AB - Superconductivity has many important applications ranging from levitating trains over qubits to MRI scanners. The phenomenon is successfully modeled by Bardeen-Cooper-Schrieffer (BCS) theory. From a mathematical perspective, BCS theory has been studied extensively for systems without boundary. However, little is known in the presence of boundaries. With the help of numerical methods physicists observed that the critical temperature may increase in the presence of a boundary. The goal of this thesis is to understand the influence of boundaries on the critical temperature in BCS theory and to give a first rigorous justification of these observations. On the way, we also study two-body Schrödinger operators on domains with boundaries and prove additional results for superconductors without boundary. BCS theory is based on a non-linear functional, where the minimizer indicates whether the system is superconducting or in the normal, non-superconducting state. By considering the Hessian of the BCS functional at the normal state, one can analyze whether the normal state is possibly a minimum of the BCS functional and estimate the critical temperature. The Hessian turns out to be a linear operator resembling a Schrödinger operator for two interacting particles, but with more complicated kinetic energy. As a first step, we study the two-body Schrödinger operator in the presence of boundaries. For Neumann boundary conditions, we prove that the addition of a boundary can create new eigenvalues, which correspond to the two particles forming a bound state close to the boundary. Second, we need to understand superconductivity in the translation invariant setting. While in three dimensions this has been extensively studied, there is no mathematical literature for the one and two dimensional cases. In dimensions one and two, we compute the weak coupling asymptotics of the critical temperature and the energy gap in the translation invariant setting. We also prove that their ratio is independent of the microscopic details of the model in the weak coupling limit; this property is referred to as universality. In the third part, we study the critical temperature of superconductors in the presence of boundaries. We start by considering the one-dimensional case of a half-line with contact interaction. Then, we generalize the results to generic interactions and half-spaces in one, two and three dimensions. Finally, we compare the critical temperature of a quarter space in two dimensions to the critical temperatures of a half-space and of the full space. AU - Roos, Barbara ID - 14374 SN - 2663 - 337X TI - Boundary superconductivity in BCS theory ER - TY - JOUR AB - We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg–Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity. AU - Hainzl, Christian AU - Roos, Barbara AU - Seiringer, Robert ID - 13207 IS - 4 JF - Journal of Spectral Theory SN - 1664-039X TI - Boundary superconductivity in the BCS model VL - 12 ER - TY - JOUR AB - 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. AU - Brooks, Morris AU - Seiringer, Robert ID - 14441 JF - Communications in Mathematical Physics SN - 0010-3616 TI - The Fröhlich Polaron at strong coupling: Part I - The quantum correction to the classical energy VL - 404 ER - TY - JOUR AB - 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. AU - Mitrouskas, David Johannes AU - Mysliwy, Krzysztof AU - Seiringer, Robert ID - 13178 JF - Forum of Mathematics TI - Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron VL - 11 ER - TY - DATA AB - 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. AU - Klausen, Frederik Ravn AU - Lauritsen, Asbjørn Bækgaard ID - 12869 TI - Research data for: A stochastic cellular automaton model of culture formation ER - TY - JOUR AB - 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. AU - Klausen, Frederik Ravn AU - Lauritsen, Asbjørn Bækgaard ID - 12890 IS - 5 JF - Physical Review E SN - 2470-0045 TI - Stochastic cellular automaton model of culture formation VL - 108 ER - TY - JOUR AB - 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. AU - Feliciangeli, Dario AU - Gerolin, Augusto AU - Portinale, Lorenzo ID - 12911 IS - 4 JF - Journal of Functional Analysis SN - 0022-1236 TI - A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature VL - 285 ER -