Sun, Xiaoshu;
(2024)
The numerical computation of Casimir energies and related spectral problems.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
Computing the Casimir force and energy between objects is a classical problem of quantum theory going back to the 1940s. Since then, various approaches have been developed based on different physical principles, such as the zeta function regularizations, the stress tensors, and the determinant of boundary layer operators. Most notably, the representation of the Casimir energy in terms of determinants of boundary layer operators makes it accessible to efficient numerical approaches. Yet, the derivation remained non-rigorous. Only recently the mathematical equivalence between these approaches are fully established. In addition, a full mathematical justification of the determinant formulae as the trace of a linear combination of powers of Laplace operators describing the Casimir energy was achieved for both the acoustic and the electromagnetic (EM) field. In this thesis, we build on previous numerical frameworks by looking into the details of the Casimir energy integral. We start by giving an overview of the relative trace formula that derives the Casimir energy formula as a representation of the integral of the log determinant of boundary integral operators. Afterwards, we validate the integrand function of Casimir energy by using it to compute the relative Krein spectral shift function within the relative trace formula and comparing it with the one computed by scattering matrices. Having shown the validation of the integrand function of Casimir energy, we present the numerical framework for computing the Casimir energy for both acoustic and EM cases in detail such as the property of the integrand function, and compare our estimates with reference values from the literature. Finally, we improve this numerical scheme with iterative methods, which are based on the spectral properties of the block matrices for the boundary layer operators. These methods allow for Casimir energy calculation for large-scale practical problems and significantly speed up computations in that case.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | The numerical computation of Casimir energies and related spectral problems |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2024. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10185254 |
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