@article{19642, abstract = {We study the criticality and subcriticality of powers (−Δ) α with α>0 of the discrete Laplacian −Δ acting on ℓ 2 (N). We prove that these positive powers of the Laplacian are critical if and only if α≥3/2. We complement our analysis with Hardy-type inequalities for (−Δ) α in the subcritical regimes α∈(0,3/2). As an illustration of the critical case α≥3/2, we analyze asymptotic properties of discrete eigenvalues emerging by coupling (−Δ) α with a localized potential.}, author = {Gerhát, Borbála M and Krejčiřík, David and Štampach, František}, issn = {2235-0616}, journal = {Revista Matematica Iberoamericana}, number = {3}, pages = {1173--1200}, publisher = {EMS Press}, title = {{Criticality transition for positive powers of the discrete Laplacian on the half line}}, doi = {10.4171/RMI/1523}, volume = {41}, year = {2025}, } @article{19660, abstract = {We analyze the ground state energy of N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We show that for weak coupling, the ground state energy is asymptotically described by the polaron energy, as proposed by F. Chevy in the physics literature. The polaron energy is the solution of a nonlinear equation involving the Green’s function of the free Fermi gas and the binding energy of the two-body point interaction. We provide quantitative error estimates that are uniform in the thermodynamic limit.}, author = {Mitrouskas, David Johannes}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {3}, publisher = {Springer Nature}, title = {{The weakly coupled two-dimensional Fermi polaron}}, doi = {10.1007/s00205-025-02098-9}, volume = {249}, year = {2025}, } @article{19661, abstract = {The Nelson model describes non-relativistic particles coupled to a relativistic Bose scalar field. In this article, we study the renormalized version of the Nelson model with massless bosons in Davies' weak coupling limit. Our main result states that the two-body Coulomb potential emerges as an effective pair interaction between the particles, which arises from the exchange of virtual excitations of the quantum field.}, author = {Cárdenas, Esteban and Mitrouskas, David Johannes}, issn = {1751-8121}, journal = {Journal of Physics A: Mathematical and Theoretical}, number = {17}, publisher = {IOP Publishing}, title = {{The renormalized Nelson model in the weak coupling limit}}, doi = {10.1088/1751-8121/adcdd9}, volume = {58}, year = {2025}, } @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{20045, abstract = {We consider the time evolution of the renormalized Nelson model, which describes N bosons linearly coupled to a quantized scalar field, in the mean-field limit of many particles N≫1 with coupling constant proportional to N^−1/2. First, we show that initial states exhibiting Bose–Einstein condensation for the particles and approximating a coherent state for the quantum field retain their structure under the many-body time evolution. Concretely, the dynamics of the reduced densities are approximated by solutions of two coupled PDEs, the Schrödinger–Klein–Gordon equations. Second, we construct a renormalized Bogoliubov evolution that describes the quantum fluctuations around the Schrödinger–Klein–Gordon equations. This evolution is used to extend the approximation of the evolved many-body state to the full norm topology. In summary, we provide a comprehensive analysis of the Nelson model that reveals the role of renormalization in the mean-field Bogoliubov theory.}, author = {Falconi, Marco and Lampart, Jonas and Leopold, Nikolai and Mitrouskas, David Johannes}, issn = {1873-1430}, journal = {Annales de l'Institut Henri Poincaré C}, publisher = {EMS Press}, title = {{Renormalized Bogoliubov theory for the Nelson model}}, doi = {10.4171/aihpc/154}, year = {2025}, } @article{20251, abstract = {The Lane–Emden inequality controls (math. formular) in terms of the L^1 and L^p norms of p. We provide a remainder estimate for this inequality in terms of a suitable distance of p to the manifold of optimizers.}, author = {Carlen, Eric and Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {7}, publisher = {Springer Nature}, title = {{Stability estimate for the Lane–Emden inequality}}, doi = {10.1007/s00526-025-03062-x}, volume = {64}, year = {2025}, } @article{20495, abstract = {We consider a tracer particle coupled to a Bose scalar field and study the regime where the field’s propagation speed approaches infinity. For initial states devoid of field excitations, we introduce an effective approximation of the time-evolved wave function and prove its validity in Hilbert space norm. In this approximation, the field remains in the vacuum state, while the tracer particle propagates with a modified dispersion relation. Physically, the new dispersion relation can be understood as the effect of radiative corrections due to interactions with virtual bosons. Mathematically, it is defined as the solution of a self-consistent nonlinear equation, whose form depends on the relevant time scale.}, author = {Cárdenas, Esteban and Mitrouskas, David Johannes}, issn = {1424-0637}, journal = {Annales Henri Poincare}, publisher = {Springer Nature}, title = {{Radiative corrections to the dynamics of a tracer particle coupled to a Bose ccalar field}}, doi = {10.1007/s00023-025-01626-3}, year = {2025}, } @article{19548, abstract = {We consider the BCS energy gap „.T / (essentially given by „.T /  .T; p/, the BCS order parameter) at all temperatures 0  T  Tc up to the critical one, Tc, and show that, in the limit of weak coupling, the ratio „.T /=Tc is given by a universal function of the relative temperature T =Tc. On the one hand, this recovers a recent result by Langmann and Triola [Phys. Rev. B 108 (2023), no. 10, article no. 104503] on three-dimensional s-wave superconductors for temperatures bounded uniformly away from Tc. On the other hand, our result lifts these restrictions, as we consider arbitrary spatial dimensions d 2 ¹1; 2; 3º, discuss superconductors with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of temperatures close to Tc. ​ .}, author = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard}, issn = {1664-0403}, journal = {Journal of Spectral Theory}, number = {1}, pages = {305–352}, publisher = {EMS Press}, title = {{Universal behavior of the BCS energy gap}}, doi = {10.4171/JST/540}, volume = {15}, year = {2025}, } @article{19372, abstract = {We consider the confined Fröhlich polaron and establish an asymptotic series for the low-energy eigenvalues in negative powers of the coupling constant. The coefficients of the series are derived through a two-fold perturbation approach, involving expansions around the electron Pekar minimizer and the excitations of the quantum field.}, author = {Brooks, Morris and Mitrouskas, David Johannes}, issn = {2690-1005}, journal = {Probability and Mathematical Physics}, number = {1}, pages = {281--325}, publisher = {Mathematical Sciences Publishers}, title = {{Asymptotic series for low-energy excitations of the Fröhlich polaron at strong coupling}}, doi = {10.2140/pmp.2025.6.281}, volume = {6}, year = {2025}, } @article{18074, abstract = {The Aharonov–Casher theorem is a result on the number of the so-called zero modes of a system described by the magnetic Pauli operator in R2. In this paper we address the same question for the Dirac operator on a flat two-dimensional manifold with boundary and Atiyah–Patodi–Singer boundary condition. More concretely we are interested in the plane and a disc with a finite number of circular holes cut out. We consider a smooth compactly supported magnetic field on the manifold and an arbitrary magnetic field inside the holes.}, author = {Fialova, Marie}, issn = {1424-0637}, journal = {Annales Henri Poincare}, pages = {2859--2900}, publisher = {Springer Nature}, title = {{Aharonov–Casher theorems for Dirac operators on manifolds with boundary and APS boundary condition}}, doi = {10.1007/s00023-024-01482-7}, volume = {26}, year = {2025}, } @article{17240, abstract = {We prove an upper bound on the energy density of the dilute spin-\(\frac {1}{2}\) Fermi gas capturing the leading correction to the kinetic energy\(8\pi a\rho _\uparrow\rho _\downarrow\) with an error of size smaller than\(a\rho^{2}(a^ 3\rho)^{1/3-\varepsilon}\) for any\(\varepsilon> 0\), where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3).}, author = {Lauritsen, Asbjørn Bækgaard}, issn = {1424-0637}, journal = {Annales Henri Poincare}, pages = {203--243}, publisher = {Springer Nature}, title = {{Almost optimal upper bound for the ground state energy of a dilute Fermi gas via cluster expansion}}, doi = {10.1007/s00023-024-01450-1}, volume = {26}, year = {2025}, } @unpublished{19550, abstract = {We introduce a multi-band BCS free energy functional and prove that for a multi-band superconductor the effect of inter-band coupling can only increase the critical temperature, irrespective of its attractive or repulsive nature and its strength. Further, for weak coupling and weaker inter-band coupling, we prove that the dependence of the increase in critical temperature on the inter-band coupling is (1) linear, if there are two or more equally strongly superconducting bands, or (2) quadratic, if there is only one dominating band.}, author = {Henheik, Sven Joscha and Langmann, Edwin and Lauritsen, Asbjørn Bækgaard}, booktitle = {arXiv}, title = {{Multi-band superconductors have enhanced critical temperatures}}, doi = {10.48550/arXiv.2409.17297}, year = {2024}, } @article{15318, abstract = {We consider a gas of N weakly interacting bosons in the ground state. Such gases exhibit Bose–Einstein condensation. The binding energy is defined as the energy it takes to remove one particle from the gas. In this article, we prove an asymptotic expansion for the binding energy, and compute the first orders explicitly for the homogeneous gas. Our result addresses in particular a conjecture by Nam (Lett Math Phys 108(1):141–159, 2018), and provides an asymptotic expansion of the ionization energy of bosonic atoms.}, author = {Bossmann, Lea and Leopold, Nikolai K and Mitrouskas, David Johannes and Petrat, Sören P}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, number = {4}, publisher = {Springer Nature}, title = {{A note on the binding energy for Bosons in the mean-field limit}}, doi = {10.1007/s10955-024-03260-5}, volume = {191}, year = {2024}, } @article{17478, abstract = {We study the Fröhlich polaron model in R3, and prove a lower bound on its ground state energy as a function of the total momentum. The bound is asymptotically sharp at large coupling. In combination with a corresponding upper bound proved earlier (Mitrouskas et al. in Forum Math. Sigma 11:1–52, 2023), it shows that the energy is approximately parabolic below the continuum threshold, and that the polaron’s effective mass (defined as the semi-latus rectum of the parabola) is given by the celebrated Landau–Pekar formula. In particular, it diverges as α4 for large coupling constant α.}, author = {Brooks, Morris and Seiringer, Robert}, issn = {1618-1913}, journal = {Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques}, pages = {271--309}, publisher = {Springer Nature}, title = {{The Fröhlich polaron at strong coupling: Part II — Energy-momentum relation and effective mass}}, doi = {10.1007/s10240-024-00150-0}, volume = {140}, year = {2024}, } @inbook{18948, abstract = {We consider a gas of N bosons with interactions in the mean-field scaling regime. We review a recent proof of the asymptotic expansion of its spectrum and eigenstates and two applications of this result, namely the derivation of an Edgeworth expansion for fluctuations of one-body operators and the computation of the binding energy of an inhomogeneous Bose gas to any order. Finally, we collect related results for the dynamics of the weakly interacting Bose gas and for the regularized Nelson model.}, author = {Bossmann, Lea and Leopold, Nikolai and Mitrouskas, David Johannes and Petrat, Sören}, booktitle = {Physics and the Nature of Reality}, editor = {Bassi, Angelo and Goldstein, Sheldon and Tumulka, Roderich and Zanghi, Nino}, isbn = {9783031454332}, issn = {2365-6425}, pages = {307--321}, publisher = {Springer Nature}, title = {{Asymptotic Analysis of the Weakly Interacting Bose Gas: A Collection of Recent Results and Applications}}, doi = {10.1007/978-3-031-45434-9_22}, volume = {215}, year = {2024}, } @article{18958, abstract = {This workshop brought together experts on the analysis of quantum many-body problems and quantum statistical mechanics, with the goal of discussing the state-of-the-art of the field, recent developments as well as challenges for the future. The main topics of discussion concerned the equilibrium and dynamical behavior of (bosonic or fermionic) quantum gases, quantum spin systems, as well as quantum field theory models like the Nelson or Fröhlich model.}, author = {Hainzl, Christian and Schlein, Benjamin and Seiringer, Robert and Warzel, Simone}, issn = {1660-8941}, journal = {Oberwolfach Reports}, number = {3}, pages = {2247--2302}, publisher = {EMS Press}, title = {{Many-body quantum systems}}, doi = {10.4171/owr/2023/39}, volume = {20}, year = {2024}, } @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{18107, abstract = {We consider a dilute fully spin-polarized Fermi gas at positive temperature in dimensions d∈{1,2,3} . We show that the pressure of the interacting gas is bounded from below by that of the free gas plus, to leading order, an explicit term of order adρ2+2/d, where a is the p-wave scattering length of the repulsive interaction and ρ is the particle density. The results are valid for a wide range of repulsive interactions, including that of a hard core, and uniform in temperatures at most of the order of the Fermi temperature. A central ingredient in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237–260).}, author = {Lauritsen, Asbjørn Bækgaard and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Pressure of a dilute spin-polarized Fermi gas: Lower bound}}, doi = {10.1017/fms.2024.56}, volume = {12}, year = {2024}, } @phdthesis{18135, abstract = {This thesis consists of two separate parts. In the first part we consider a dilute Fermi gas interacting through a repulsive interaction in dimensions $d=1,2,3$. Our focus is mostly on the physically most relevant dimension $d=3$ and the setting of a spin-polarized (equivalently spinless) gas, where the Pauli exclusion principle plays a key role. We show that, at zero temperature, the ground state energy density of the interacting spin-polarized gas differs (to leading order) from that of the free (i.e. non-interacting) gas by a term of order $a_p^d\rho^{2+2/d}$ with $a_p$ the $p$-wave scattering length of the repulsive interaction and $\rho$ the density. Further, we extend this to positive temperature and show that the pressure of an interacting spin-polarized gas differs from that of the free gas by a now temperature dependent term, again of order $a_p^d\rho^{2+2/d}$. Lastly, we consider the setting of a spin-$\frac{1}{2}$ Fermi gas in $d=3$ dimensions and show that here, as an upper bound, the ground state energy density differs from that of the free system by a term of order $a_s \rho^2$ with an error smaller than $a_s \rho^2 (a_s\rho^{1/3})^{1-\eps}$ for any $\eps > 0$, where $a_s$ is the $s$-wave scattering length of the repulsive interaction. These asymptotic formulas complement the similar formulas in the literature for the dilute Bose and spin-$\frac{1}{2}$ Fermi gas, where the ground state energies or pressures differ from that of the corresponding free systems by a term of order $a_s \rho^2$ in dimension $d=3$. In the spin-polarized setting, the corrections, of order $a_p^3\rho^{8/3}$ in dimension $d=3$, are thus much smaller and requires a more delicate analysis. In the second part of the thesis we consider the Bardeen--Cooper--Schrieffer (BCS) theory of superconductivity and in particular its associated critical temperature and energy gap. We prove that the ratio of the zero-temperature energy gap and critical temperature $\Xi(T=0)/T_c$ approaches a universal constant $\pi e^{-\gamma}\approx 1.76$ in both the limit of high density in dimension $d=3$ and in the limit of weak coupling in dimensions $d=1,2$. This complements the proofs in the literature of this universal behaviour in the limit of weak coupling or low density in dimension $d=3$. Secondly, we prove that the ratio of the energy gap at positive temperature and critical temperature $\Xi(T)/T_c$ approaches a universal function of the relative temperature $T/T_c$ in the limit of weak coupling in dimensions $d=1,2,3$.}, author = {Lauritsen, Asbjørn Bækgaard}, isbn = {978-3-99078-042-8}, issn = {2663-337X}, pages = {353}, publisher = {Institute of Science and Technology Austria}, title = {{Energies of dilute Fermi gases and universalities in BCS theory}}, doi = {10.15479/at:ista:18135}, year = {2024}, } @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}, }