@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},
}

@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{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},
}