---
res:
  bibo_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 $d=3$ \r\nand 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. \r\n\r\nThese
    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.\r\n\r\nIn 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 $d=1,2,3$.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Asbjørn Bækgaard
      foaf_name: Lauritsen, Asbjørn Bækgaard
      foaf_surname: Lauritsen
      foaf_workInfoHomepage: http://www.librecat.org/personId=e1a2682f-dc8d-11ea-abe3-81da9ac728f1
    orcid: 0000-0003-4476-2288
  bibo_doi: 10.15479/at:ista:18135
  dct_date: 2024^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/2663-337X
  - http://id.crossref.org/issn/978-3-99078-042-8
  dct_language: eng
  dct_publisher: Institute of Science and Technology Austria@
  dct_title: Energies of dilute Fermi gases and universalities in BCS theory@
...