Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.785109
Title: Szegö-type trace asymptotics for operators with translational symmetry
Author: Pfirsch, Bernhard
ISNI:       0000 0004 7970 6523
Awarding Body: UCL (University College London)
Current Institution: University College London (University of London)
Date of Award: 2019
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Abstract:
The classical Szegö 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.
Supervisor: Parnovski, L. ; Sobolev, A. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.785109  DOI: Not available
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