Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra

TL;DR

This paper investigates the pseudospectra of the complex harmonic oscillator using two techniques: JWKB method for resolvent estimates and analysis of the semigroup's Hilbert-Schmidt properties, extending classical formulas.

Contribution

It introduces new resolvent norm estimates and demonstrates the Hilbert-Schmidt nature of the semigroup generated by the non-self-adjoint harmonic oscillator.

Findings

Resolved resolvent norm estimates using JWKB method.

Showed the semigroup is of Hilbert-Schmidt type in a maximal angular region.

Extended classical Mehler's formula to a non-self-adjoint setting.

Abstract

We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of Hilbert-Schmidt type in a maximal angular region. In order to show this last property, we deduce a non-self-adjoint version of the classical Mehler's formula.