@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}, } @phdthesis{11473, abstract = {The polaron model is a basic model of quantum field theory describing a single particle interacting with a bosonic field. It arises in many physical contexts. We are mostly concerned with models applicable in the context of an impurity atom in a Bose-Einstein condensate as well as the problem of electrons moving in polar crystals. The model has a simple structure in which the interaction of the particle with the field is given by a term linear in the field’s creation and annihilation operators. In this work, we investigate the properties of this model by providing rigorous estimates on various energies relevant to the problem. The estimates are obtained, for the most part, by suitable operator techniques which constitute the principal mathematical substance of the thesis. The first application of these techniques is to derive the polaron model rigorously from first principles, i.e., from a full microscopic quantum-mechanical many-body problem involving an impurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas in the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak interactions as a low-energy effective theory for this problem. In the second part, we investigate rigorously the ground state of the model at fixed momentum and for large values of the coupling constant. Qualitatively, the system is expected to display a transition from the quasi-particle behavior at small momenta, where the dispersion relation is parabolic and the particle moves through the medium dragging along a cloud of phonons, to the radiative behavior at larger momenta where the polaron decelerates and emits free phonons. At the same time, in the strong coupling regime, the bosonic field is expected to behave purely classically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to be asymptotically equal to the one obtained from the semiclassical counterpart of the problem, first studied by Landau and Pekar in the 1940s. For polaron models with regularized form factors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear function of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove that for a large window of momenta below the radiation threshold, the energy-momentum relation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the Landau–Pekar effective mass, as expected. For the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is of the optical type and the form factor is formally UV–singular due to the nature of the point charge-dipole interaction, we are able to give the corresponding upper bound. In contrast to the regular case, this requires the inclusion of the quantum fluctuations of the phonon field, which makes the problem considerably more difficult. The results are supplemented by studies on the absolute ground-state energy at strong coupling, a proof of the divergence of the effective mass with the coupling constant for a wide class of polaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model and the application of the techniques used for its removal for the energy estimates. }, author = {Mysliwy, Krzysztof}, issn = {2663-337X}, pages = {138}, publisher = {Institute of Science and Technology Austria}, title = {{Polarons in Bose gases and polar crystals: Some rigorous energy estimates}}, doi = {10.15479/at:ista:11473}, 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{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{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}, }