Algebras of semiclassical pseudodifferential operators associated with Zoll-type domains in cotangent bundles

TL;DR

This paper develops an algebra of semiclassical pseudodifferential operators for Zoll-type domains in cotangent bundles, providing symbolic calculus, quantization, and limit theorems relevant to geometric analysis.

Contribution

It introduces a new algebra of operators with singular symbols for Zoll-type domains, including symbolic calculus and quantization methods.

Findings

Constructed an algebra of operators with singular symbols.

Proved the existence of projectors quantizing the domain.

Provided a symbolic proof of a Szeg"o limit theorem.

Abstract

We are consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a Bohr-Sommerfeld condition (for example, the unit disk bundle of a Zoll metric). Given such a domain, we construct an algebra of associated semiclassical pseudodifferential operators with singular symbols. The Schwartz kernels of the operators have frequency set contained in the union of the diagonal and the flow-out of the null foliation of the boundary of the domain. We develop a symbolic calculus, prove the existence of projectors (under a mild additional assumption) whose range can be thought of as quantizing the domain, give a symbolic proof of a Szeg\"o limit theorem, and study associated propagators.