@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{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}, } @phdthesis{14374, abstract = {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.}, author = {Roos, Barbara}, issn = {2663 - 337X}, pages = {206}, publisher = {Institute of Science and Technology Austria}, title = {{Boundary superconductivity in BCS theory}}, doi = {10.15479/at:ista:14374}, year = {2023}, } @article{13207, abstract = {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.}, author = {Hainzl, Christian and Roos, Barbara and Seiringer, Robert}, issn = {1664-0403}, journal = {Journal of Spectral Theory}, number = {4}, pages = {1507–1540}, publisher = {EMS Press}, title = {{Boundary superconductivity in the BCS model}}, doi = {10.4171/JST/439}, volume = {12}, 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{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}, } @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{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}, publisher = {World Scientific Publishing}, title = {{Universality in low-dimensional BCS theory}}, doi = {10.1142/s0129055x2360005x}, 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{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{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{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{14854, abstract = { 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}, } @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}, } @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{12276, abstract = {Ongoing development of quantum simulators allows for a progressively finer degree of control of quantum many-body systems. This motivates the development of efficient approaches to facilitate the control of such systems and enable the preparation of nontrivial quantum states. Here we formulate an approach to control quantum systems based on matrix product states (MPSs). We compare counterdiabatic and leakage minimization approaches to the so-called local steering problem that consists in finding the best value of the control parameters for generating a unitary evolution of the specific MPS in a given direction. In order to benchmark the different approaches, we apply them to the generalization of the PXP model known to exhibit coherent quantum dynamics due to quantum many-body scars. We find that the leakage-based approach generally outperforms the counterdiabatic framework and use it to construct a Floquet model with quantum scars. We perform the first steps towards global trajectory optimization and demonstrate entanglement steering capabilities in the generalized PXP model. Finally, we apply our leakage minimization approach to construct quantum scars in the periodically driven nonintegrable Ising model.}, author = {Ljubotina, Marko and Roos, Barbara and Abanin, Dmitry A. and Serbyn, Maksym}, issn = {2691-3399}, journal = {PRX Quantum}, keywords = {General Medicine}, number = {3}, publisher = {American Physical Society}, title = {{Optimal steering of matrix product states and quantum many-body scars}}, doi = {10.1103/prxquantum.3.030343}, volume = {3}, year = {2022}, }