Wodzicki residue for operators on manifolds with cylindrical ends

Abstract

Abstract. We define the Wodzicki Residue TR(A) for A belonging to a space of operators with double order, denoted L m 1 ,m 2 cl . 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 ζ(A, z). Using this approach, under suitable ellipticity assumptions, we also compute a two terms leading part of the Weyl formula for a positive selfadjoint operator A ∈ L m 1 ,m 2 cl in the case m 1 = m 2 .

A note on the Einstein–Hilbert action and Dirac operators on $${\mathbb{R}^n}$$

“…The method above can be used to treat also the case of manifolds with cylindrical ends, using the contents of [8]: one defines in this setting a regularised Wodzicki residue and exploits its connection with the zeta function. The asymptotic expansion of the heat kernel as t 0 is locally defined, so, using suitable regularised integrals, see [8], the results can be generalised to those manifolds in this class which admit a spin structure.…”

“…The definition of A z is then extended to arbitrary z ∈ C in the standard way, that is A z := A z− j • A j , where j ∈ Z + is chosen so that Re(z) − j < 0, see, e.g., [8,12,26,32,34].…”

“…Wang [35][36][37] suggested an extension of the result to a class of manifolds with boundary. In the last years, the definition of the Wodzicki residue has been extended to other settings: manifolds with boundary [14], manifolds with conical singularities [15,33], SG-calculus on R n [28], anisotropic operators on R n [9], and manifolds with cylindrical ends [8]. In [26], in order to study critical metrics on R n , it has been introduced the regularised trace of operators, which leads to the definition of regularised ζ -function.…”

Sharp Weyl estimates for tensor products of pseudodifferential operators

“…However, it was not possible, through the aforementioned method, to give a good estimate of the remainder term. We notice that the asymptotic behavior of the counting function in the bisingular case has some similarities with the Weyl law in the setting of SG-classical operators on manifolds with ends [BC11,CM13].…”

“…Similar formulae can be obtained in many other different settings, see [SV97] and [ANPS09] for a detailed analysis and several developments. To mention a few specific situations, see [Shu87,HR81] for the case of the Shubin calculus on R n , [BN03] for the anisotropic Shubin calculus, [BC11,CM13,Nic03] for the SG-operators on R n and the manifolds with ends, [GL02] for operators on conic manifolds, [Mor08] for operators 1 on cusp manifolds, [DD13] for operators on asymptotic hyperbolic manifolds, [Bat12,BGRP13] for bisingular operators.…”