@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{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{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}, } @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{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{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{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{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{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{10755, abstract = {We provide a definition of the effective mass for the classical polaron described by the Landau–Pekar (LP) equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by LP (1948 J. Exp. Theor. Phys. 18 419–423).}, author = {Feliciangeli, Dario and Rademacher, Simone Anna Elvira and Seiringer, Robert}, issn = {1751-8121}, journal = {Journal of Physics A: Mathematical and Theoretical}, number = {1}, publisher = {IOP Publishing}, title = {{The effective mass problem for the Landau-Pekar equations}}, doi = {10.1088/1751-8121/ac3947}, volume = {55}, year = {2022}, } @article{10850, abstract = {We study two interacting quantum particles forming a bound state in d-dimensional free space, and constrain the particles in k directions to (0, ∞)k ×Rd−k, with Neumann boundary conditions. First, we prove that the ground state energy strictly decreases upon going from k to k+1. This shows that the particles stick to the corner where all boundary planes intersect. Second, we show that for all k the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413–1444, 2020) to dimensions d > 1.}, author = {Roos, Barbara and Seiringer, Robert}, issn = {0022-1236}, journal = {Journal of Functional Analysis}, keywords = {Analysis}, number = {12}, publisher = {Elsevier}, title = {{Two-particle bound states at interfaces and corners}}, doi = {10.1016/j.jfa.2022.109455}, volume = {282}, year = {2022}, } @article{10564, abstract = {We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau–Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass.}, author = {Mysliwy, Krzysztof and Seiringer, Robert}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, number = {1}, publisher = {Springer Nature}, title = {{Polaron models with regular interactions at strong coupling}}, doi = {10.1007/s10955-021-02851-w}, volume = {186}, year = {2022}, } @article{11917, abstract = {We study the many-body dynamics of an initially factorized bosonic wave function in the mean-field regime. We prove large deviation estimates for the fluctuations around the condensate. We derive an upper bound extending a recent result to more general interactions. Furthermore, we derive a new lower bound which agrees with the upper bound in leading order.}, author = {Rademacher, Simone Anna Elvira and Seiringer, Robert}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, publisher = {Springer Nature}, title = {{Large deviation estimates for weakly interacting bosons}}, doi = {10.1007/s10955-022-02940-4}, volume = {188}, year = {2022}, } @article{12246, abstract = {The Lieb–Oxford inequality provides a lower bound on the Coulomb energy of a classical system of N identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree–Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy.}, author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {5}, publisher = {Springer Nature}, title = {{Improved Lieb–Oxford bound on the indirect and exchange energies}}, doi = {10.1007/s11005-022-01584-5}, volume = {112}, year = {2022}, } @article{10738, abstract = {We prove an adiabatic theorem for the Landau–Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fröhlich Hamiltonian with large coupling constant α. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau–Pekar equations until times short compared to α2.}, author = {Leopold, Nikolai K and Rademacher, Simone Anna Elvira and Schlein, Benjamin and Seiringer, Robert}, issn = {1948-206X}, journal = {Analysis and PDE}, number = {7}, pages = {2079--2100}, publisher = {Mathematical Sciences Publishers}, title = {{ The Landau–Pekar equations: Adiabatic theorem and accuracy}}, doi = {10.2140/APDE.2021.14.2079}, volume = {14}, year = {2021}, } @article{10852, abstract = { We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass.}, author = {Seiringer, Robert}, issn = {1793-6659}, journal = {Reviews in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {01}, publisher = {World Scientific Publishing}, title = {{The polaron at strong coupling}}, doi = {10.1142/s0129055x20600120}, volume = {33}, year = {2021}, } @article{7901, abstract = {We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree–Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy.}, author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {885--979}, publisher = {Springer}, title = {{Correlation energy of a weakly interacting Fermi gas}}, doi = {10.1007/s00222-021-01041-5}, volume = {225}, year = {2021}, } @article{8603, abstract = {We consider the Fröhlich polaron model in the strong coupling limit. It is well‐known that to leading order the ground state energy is given by the (classical) Pekar energy. In this work, we establish the subleading correction, describing quantum fluctuation about the classical limit. Our proof applies to a model of a confined polaron, where both the electron and the polarization field are restricted to a set of finite volume, with linear size determined by the natural length scale of the Pekar problem.}, author = {Frank, Rupert and Seiringer, Robert}, issn = {10970312}, journal = {Communications on Pure and Applied Mathematics}, number = {3}, pages = {544--588}, publisher = {Wiley}, title = {{Quantum corrections to the Pekar asymptotics of a strongly coupled polaron}}, doi = {10.1002/cpa.21944}, volume = {74}, year = {2021}, } @article{14889, abstract = {We consider the Fröhlich Hamiltonian with large coupling constant α. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order α2. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order α2.}, author = {Leopold, Nikolai K and Mitrouskas, David Johannes and Rademacher, Simone Anna Elvira and Schlein, Benjamin and Seiringer, Robert}, issn = {2578-5885}, journal = {Pure and Applied Analysis}, number = {4}, pages = {653--676}, publisher = {Mathematical Sciences Publishers}, title = {{Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron}}, doi = {10.2140/paa.2021.3.653}, volume = {3}, year = {2021}, }