Title:

Twoterm Szegő theorem for generalised antiWick operators

This thesis concerns operators whose Weyl pseudodifferential operator symbol is the convolution of a function that is smooth and of fixed scale with a function that is discontinuous and dilated by a large asymptotic parameter. A special case of these operators of particular interest is the class of generalised antiWick operators, and then the fixedscale part corresponds to the window functions while the dilated part is the generalised antiWick symbol. The main result is a Szegő theorem that gives two terms in the asymptotic expansion of the trace of a function of the operator. Two variants are proved: in one the discontinuity must occur on a C^2 surface but the symbol may have unbounded support, while in the other the set on which the discontinuity occurs may be much more general (most importantly, it must be Lipschitz and piecewise C^2), but the symbol must be compactly supported. A corollary of this theorem is two terms in the asymptotic expansion of the eigenvalue counting function when the smooth part of the symbol is constant. Prior to this work, only one term in each of these expansions was known. It is also shown that the remainder in the Szegő theorem is larger for a class of examples where the boundary has a cusp; this shows that the Lipschitz condition in the main theorem cannot be removed without weakening the conclusion. A significant step in the proof of this Szegő theorem is a composition result for Weyl pseudodifferential operators that may be of more general interest: the symbol of the composition is expressed as a finite series in the standard form, but with an explicit trace norm and operator norm bound of the remainder expressed using the symbols in a similar way to the first excluded term. In the oneterm case, this is used to derive an analogous trace norm bound for approximating the Weyl symbol of a function of an operator. Another important part of the proof of the Szegő theorem is the use of standard tubular neighbourhood theory to describe the geometry of the surface on which the discontinuity occurs; this is derived in full for the necessary conditions.
