Determinants of Classical SG-Pseudodifferential Operators

TL;DR

This paper develops a generalized trace functional for classical SG-pseudodifferential operators, enabling the definition of zeta-regularized determinants and analyzing their asymptotics, with implications for elliptic operator theory.

Contribution

It introduces a new trace functional for SG-operators and defines their zeta-regularized determinants, extending previous frameworks to broader classes of operators.

Findings

Defined a generalized trace functional TR for SG-operators.

Established asymptotic behaviors of TR exp(-tA) and TR (-A)^{-k}.

Connected the determinant of operators to relative determinants for suitable pairs.

Abstract

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG-pseudodifferential operators on R^n and suitable manifolds, using a finite-part integral regularization technique. This allows us to define a zeta-regularized determinant det A for classical parameter-elliptic SG-operators A of order (\mu,m), with \mu>0, m\ge0. For m=0, the asymptotics of TR exp(-tA) as t\to 0 and of TR (\lambda-A)^{-k}$ as |\lambda|\to\infty are derived. For suitable pairs (A,A_0) we show that det A/det A_0 coincides with the so-called relative determinant det(A,A_0).