---
OA_place: publisher
_id: '18135'
abstract:
- lang: eng
  text: "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 $d=1,2,3$."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Asbjørn Bækgaard
  full_name: Lauritsen, Asbjørn Bækgaard
  id: e1a2682f-dc8d-11ea-abe3-81da9ac728f1
  last_name: Lauritsen
  orcid: 0000-0003-4476-2288
citation:
  ama: Lauritsen AB. Energies of dilute Fermi gases and universalities in BCS theory.
    2024. doi:<a href="https://doi.org/10.15479/at:ista:18135">10.15479/at:ista:18135</a>
  apa: Lauritsen, A. B. (2024). <i>Energies of dilute Fermi gases and universalities
    in BCS theory</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/at:ista:18135">https://doi.org/10.15479/at:ista:18135</a>
  chicago: Lauritsen, Asbjørn Bækgaard. “Energies of Dilute Fermi Gases and Universalities
    in BCS Theory.” Institute of Science and Technology Austria, 2024. <a href="https://doi.org/10.15479/at:ista:18135">https://doi.org/10.15479/at:ista:18135</a>.
  ieee: A. B. Lauritsen, “Energies of dilute Fermi gases and universalities in BCS
    theory,” Institute of Science and Technology Austria, 2024.
  ista: Lauritsen AB. 2024. Energies of dilute Fermi gases and universalities in BCS
    theory. Institute of Science and Technology Austria.
  mla: Lauritsen, Asbjørn Bækgaard. <i>Energies of Dilute Fermi Gases and Universalities
    in BCS Theory</i>. Institute of Science and Technology Austria, 2024, doi:<a href="https://doi.org/10.15479/at:ista:18135">10.15479/at:ista:18135</a>.
  short: A.B. Lauritsen, Energies of Dilute Fermi Gases and Universalities in BCS
    Theory, Institute of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-09-24T10:56:25Z
date_published: 2024-09-23T00:00:00Z
date_updated: 2026-04-16T08:17:55Z
day: '23'
ddc:
- '515'
- '539'
degree_awarded: PhD
department:
- _id: GradSch
- _id: RoSe
doi: 10.15479/at:ista:18135
ec_funded: 1
file:
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has_accepted_license: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '09'
oa: 1
oa_version: Published Version
page: '353'
project:
- _id: bda63fe5-d553-11ed-ba76-a16e3d2f256b
  grant_number: I06427
  name: Mathematical Challenges in BCS Theory of Superconductivity
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '694227'
  name: Analysis of quantum many-body systems
publication_identifier:
  isbn:
  - 978-3-99078-042-8
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
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status: public
supervisor:
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
title: Energies of dilute Fermi gases and universalities in BCS theory
tmp:
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  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2024'
...