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

Abstract

We identify the norm of the semigroup generated by the non-self-adjoint harmonic oscillator acting on $L^2(\mathbb{R})$, 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 $L^2({\mathbb{R}}^n)$. The method used --- identifying the exponents of sharp products of Mehler formulas --- is elementary and is inspired by more general works of L. Hörmander, A. Melin, and J. Sjöstrand.

22 pages, 1 figure. Significant revision following referee's comments. To appear in Integral Equations and Operator Theory; published version may differ

References in corpus (3)

• Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator
• Estimation de résolvante et construction de quasimode près du bord du pseudospectre
• Singular-value decomposition of solution operators to model evolution equations

Cited by in corpus (1)

• The elliptic evolution of non-self-adjoint degree-2 Hamiltonians