The norm of the non-self-adjoint harmonic oscillator semigroup

TL;DR

This paper determines the exact norm of the semigroup generated by the non-self-adjoint harmonic oscillator on L^2 spaces, relating it to Gaussian-weighted holomorphic function spaces and extending to elliptic quadratic operators in multiple dimensions.

Contribution

It provides a precise computation of the semigroup norm for non-self-adjoint harmonic oscillators and introduces an elementary method based on Mehler formulas applicable in any dimension.

Findings

Exact norm of the non-self-adjoint harmonic oscillator semigroup identified.

Method applies to elliptic quadratic operators in multiple dimensions.

Connections established between semigroup norms and Gaussian-weighted holomorphic spaces.

Abstract

We identify the norm of the semigroup generated by the non-self-adjoint harmonic oscillator acting on $L^2(\Bbb{R})$, for all complex times where it is bounded. We relate this problem to embeddings between Gaussian-weighted spaces of holomorphic functions, and we show that the same technique applies, in any dimension, to the semigroup $e^{-tQ}$ generated by an elliptic quadratic operator acting on $L^2(\Bbb{R}^n)$. The method used --- identifying the exponents of sharp products of Mehler formulas --- is elementary and is inspired by more general works of L. H\"ormander, A. Melin, and J. Sj\"ostrand.