Wodzicki Residue for Operators on Manifolds with Cylindrical Ends

TL;DR

This paper extends the concept of Wodzicki Residue to operators on manifolds with cylindrical ends, providing a new analytical framework and computing leading terms of the Weyl formula under ellipticity conditions.

Contribution

It introduces a definition of Wodzicki Residue for double order operators on non-compact manifolds with cylindrical ends and computes the leading Weyl term for certain elliptic operators.

Findings

Defined Wodzicki Residue for operators on manifolds with cylindrical ends.

Computed the leading part of the Weyl formula for positive selfadjoint operators.

Extended residue theory to a class of non-compact manifolds.

Abstract

We define the Wodzicki Residue TR(A) for A in a space of operators with double order (m_1,m_2). Such operators are globally defined initially on R^n and then, more generally, on a class of non-compact manifolds, namely, the manifolds with cylindrical ends. The definition is based on the analysis of the associate zeta function. Using this approach, under suitable ellipticity assumptions, we also compute a two terms leading part of the Weyl formula for a positive selfadjoint operator belonging the mentioned class in the case m_1=m_2.