Resolvent trace asymptotics for operators in the Shubin class

TL;DR

The paper develops a unified Shubin-type pseudodifferential calculus that simultaneously handles parameter-dependent and independent operators on $\mathbb{R}^n$. It introduces several refined symbol classes with regularity number and expansion-at-infinity structures, defines multiple principal symbols (homogeneous, limit, angular), and proves composition, adjoint, and ellipticity results, along with parametrix construction. A key achievement is linking the new calculus to Grubb–Seeley resolvent theory, yielding resolvent trace expansions that determine zeta-function poles and heat-kernel asymptotics. The main outcomes include explicit resolvent expansions for elliptic Shubin-type operators and general trace expansions for a broad class of symbols, with precise dependence on degrees and regularity.

Abstract

A new pseudodifferential calculus of Shubin type is introduced. The calculus contains operators depending on a non negative real parameter as well as operators independent of the parameter. Resolvents of Shubin type pseudodifferential operators are constructed and their trace expansion is obtained.

Figures (1)

• Figure 1: The punctured sphere $\widehat{{\mathbb S}}^m_+$ and the manifold $\mathbb{B}$

Theorems & Definitions (38)

• Remark 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Definition 2.4
• Lemma 2.5
• Proposition 2.6
• Corollary 2.7
• proof
• Definition 2.8