@article{19705, abstract = {A maximal realization of the two-dimensional Pauli operator, subject to Aharonov–Bohm magnetic field, is investigated. Contrary to the case of the Pauli operator with regular magnetic potentials, it is shown that both components of the Pauli operator are critical. Asymptotics of the weakly coupled eigenvalues, generated by electric (not necessarily self-adjoint) perturbations, are derived.}, author = {Fialova, Marie and Krejčiřík, David}, issn = {1793-6659}, journal = {Reviews in Mathematical Physics}, number = {6}, publisher = {World Scientific Publishing}, title = {{Virtual bound states of the Pauli operator with an Aharonov–Bohm potential}}, doi = {10.1142/S0129055X25500114}, volume = {37}, year = {2025}, } @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{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}, } @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{7685, abstract = {We consider a gas of interacting bosons trapped in a box of side length one in the Gross–Pitaevskii limit. We review the proof of the validity of Bogoliubov’s prediction for the ground state energy and the low-energy excitation spectrum. This note is based on joint work with C. Brennecke, S. Cenatiempo and B. Schlein.}, author = {Boccato, Chiara}, issn = {0129-055X}, journal = {Reviews in Mathematical Physics}, number = {1}, publisher = {World Scientific Publishing}, title = {{The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime}}, doi = {10.1142/S0129055X20600065}, volume = {33}, year = {2021}, } @article{7900, abstract = {Hartree–Fock theory has been justified as a mean-field approximation for fermionic systems. However, it suffers from some defects in predicting physical properties, making necessary a theory of quantum correlations. Recently, bosonization of many-body correlations has been rigorously justified as an upper bound on the correlation energy at high density with weak interactions. We review the bosonic approximation, deriving an effective Hamiltonian. We then show that for systems with Coulomb interaction this effective theory predicts collective excitations (plasmons) in accordance with the random phase approximation of Bohm and Pines, and with experimental observation.}, author = {Benedikter, Niels P}, issn = {1793-6659}, journal = {Reviews in Mathematical Physics}, number = {1}, publisher = {World Scientific Publishing}, title = {{Bosonic collective excitations in Fermi gases}}, doi = {10.1142/s0129055x20600090}, volume = {33}, year = {2021}, } @article{9285, abstract = {We first review the problem of a rigorous justification of Kubo’s formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo’s formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper, we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.}, author = {Henheik, Sven Joscha and Teufel, Stefan}, issn = {0129-055X}, journal = {Reviews in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {01}, publisher = {World Scientific Publishing}, title = {{Justifying Kubo’s formula for gapped systems at zero temperature: A brief review and some new results}}, doi = {10.1142/s0129055x20600041}, volume = {33}, year = {2021}, } @article{2744, abstract = {We study the long time evolution of a quantum particle interacting with a random potential in the Boltzmann-Grad low density limit. We prove that the phase space density of the quantum evolution defined through the Husimi function converges weakly to a linear Boltzmann equation. The Boltzmann collision kernel is given by the full quantum scattering cross-section of the obstacle potential.}, author = {Eng, David and Erdös, László}, issn = {1793-6659}, journal = {Reviews in Mathematical Physics}, number = {6}, pages = {669 -- 743}, publisher = {World Scientific Publishing}, title = {{The linear Boltzmann equation as the low density limit of a random Schrödinger equation}}, doi = {10.1142/S0129055X0500242X}, volume = {17}, year = {2005}, } @article{2734, abstract = {In this paper we describe an intrinsically geometric way of producing magnetic fields on S3 and R3 for which the corresponding Dirac operators have a non-trivial kernel. In many cases we are able to compute the dimension of the kernel. In particular we can give examples where the kernel has any given dimension. This generalizes the examples of Loss and Yau [1].}, author = {Erdös, László and Solovej, Jan}, issn = {0129-055X}, journal = {Reviews in Mathematical Physics}, number = {10}, pages = {1247 -- 1280}, publisher = {World Scientific Publishing}, title = {{The kernel of Dirac operators on S3 and R3}}, doi = {10.1142/S0129055X01000983}, volume = {13}, year = {2001}, }