# Quantization of Gaussians

## TL;DR

This paper studies the oscillator semigroup of operators with Gaussian kernels, providing formulas for their norms and identifying a subset with singular behavior, advancing understanding of Gaussian quantizations.

## Contribution

It introduces a Weyl symbol parametrization for the oscillator semigroup and derives explicit formulas for norms, highlighting the quantum degenerate subset.

## Key findings

- Formulas for tracial and operator norms of Gaussian quantizations
- Identification of quantum degenerate Gaussians with singular norms
- Enhanced understanding of the structure of the oscillator semigroup

## Abstract

Our paper is devoted to the oscillator semigroup, which can be defined as the set of operators whose kernels are centered Gaussian. Equivalently, they can be defined as the the Weyl quantization of centered Gaussians. We use the Weyl symbol as the main parametrization of this semigroup. We derive formulas for the tracial and operator norm of the Weyl quantization of Gaussians. We identify the subset of Gaussians, which we call quantum degenerate, where these norms have a singularity.