Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary

TL;DR

This paper investigates the asymptotic behavior of vacuum expectation values for second order differential operators on manifolds with boundary, using trace techniques for Laplace-type operators under specific boundary conditions.

Contribution

It provides new asymptotic formulas for vacuum expectation values of second order operators on manifolds with boundary, extending previous results to include boundary effects and specific boundary conditions.

Findings

Derived asymptotic formulas for vacuum expectation values

Analyzed effects of boundary conditions on operator traces

Extended existing theory to manifolds with boundary

Abstract

Let M be a compact Riemannian manifold with smooth boundary. We study the vacuum expectation value of an operator Q by studying Tr Qe^{-tD}, where D is an operator of Laplace type on M, and where Q is a second order operator with scalar leading symbol; we impose Dirichlet or modified Neumann boundary conditions.