Title:

Szegötype trace asymptotics for operators with translational symmetry

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 WienerHopf operators on intervals of growing length. We study two problems related to these scaling asymptotics. The first problem concerns the higherdimensional version of the trace asymptotics. Namely, consider a translationinvariant bounded linear operator in dimension two whose integral kernel exhibits superpolynomial offdiagonal 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 wellknown. 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 WienerHopf 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 WienerHopf 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 twoterm 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.
