An Index Formula for Groups of Isometric Linear Canonical Transformations

TL;DR

This paper develops an index formula for a class of operators generated by metaplectic and pseudodifferential operators, linking group representations to operator algebra properties in quantum mechanics.

Contribution

It introduces a new index formula for operators arising from isometric linear canonical transformations within a specific operator algebra.

Findings

Proves the Fredholm property for certain operators in the algebra.

Derives an explicit index formula for these operators.

Establishes conditions under which the index formula applies.

Abstract

We define a representation of the unitary group $U(n)$ by metaplectic operators acting on $L^2(\mathbb{R}^n)$ and consider the operator algebra generated by the operators of the representation and pseudodifferential operators of Shubin class. Under suitable conditions, we prove the Fredholm property for elements in this algebra and obtain an index formula.