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Szegő-type trace asymptotics for operators with translational symmetry

Pfirsch, Bernhard; (2019) Szegő-type trace asymptotics for operators with translational symmetry. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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The classical Szegő limit theorem describes the asymptotic behaviour of Toeplitz determinants as the size of the Toeplitz matrix grows. The continuous analogue are trace asymptotics for Wiener--Hopf operators on intervals of growing length. We study two problems related to these scaling asymptotics. The first problem concerns the higher-dimensional version of the trace asymptotics. Namely, consider a translation-invariant bounded linear operator in dimension two whose integral kernel exhibits super-polynomial off-diagonal decay. Then we study the spectral asymptotics of its spatial restriction to the interior of a scaled polygon, as the scaling parameter tends to infinity. To this end, we provide complete trace asymptotics for analytic functions of the truncated operator. These consist of three terms, which reflect the geometry of the polygon. If the polygon is substituted by a domain with smooth boundary, then the corresponding asymptotics are well-known. However, we show that the constant order term in the expansion for the polygon cannot be recovered from a formal approximation by smooth domains. This fact is reminiscent of the heat trace anomaly for the Dirichlet Laplacian. A prominent application of trace asymptotics for Wiener--Hopf operators lies in quantum information theory: they can be used to compute the bipartite entanglement entropy for the ground state of a free Fermi gas in the absence of an external field. At zero temperature, this requires studying Wiener--Hopf operators with a discontinuous symbol, which causes notable difficulties. In the second part of the thesis, based on joint work with Alexander V. Sobolev, we prove a two-term asymptotic trace formula for the periodic Schrödinger operator in dimension one. This formula can be applied to compute the aforementioned entanglement entropy when the fermions are exposed to a periodic electric field. Moreover, the subleading order of the asymptotics identifies the spectrum of the periodic Schrödinger operator.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Szegő-type trace asymptotics for operators with translational symmetry
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Copyright © The Author 2019. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) Licence (https://creativecommons.org/licenses/by-nc-nd/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/10076970
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