Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator

TL;DR

This paper investigates conditions under which perturbations of Gibbs semigroups remain Gibbs semigroups, focusing on the non-selfadjoint harmonic oscillator with potential growth, and establishes convergence criteria and spectral asymptotics.

Contribution

It provides new sufficient conditions for the preservation of Gibbs semigroup properties under perturbations and analyzes the spectral behavior of the non-selfadjoint harmonic oscillator with growing potentials.

Findings

Convergence of Dyson-Phillips expansion in Schatten norms for perturbed non-selfadjoint harmonic oscillator.

Conditions ensuring the perturbed operator generates a Gibbs semigroup.

Asymptotic eigenvalue and resolvent norm estimates for the perturbed operator.

Abstract

Let $T$ be the generator of a $C_0$-semigroup $e^{-Tt}$ which is of finite trace for all $t>0$ (a Gibbs semigroup). Let $A$ be another closed operator, $T$-bounded with $T$-bound equal to zero. In general $T+A$ might not be the generator of a Gibbs semigroup. In the first half of this paper we give sufficient conditions on $A$ so that $T+A$ is the generator of a Gibbs semigroup. We determine these conditions in terms of the convergence of the Dyson-Phillips expansion corresponding to the perturbed semigroup in suitable Schatten-von Neumann norms. In the second half of the paper we consider $T=H_\vartheta=-e^{-i\vartheta}\partial_x^2+e^{i\vartheta}x^2$, the non-selfadjoint harmonic oscillator, on $L^2(\mathbb{R})$ and $A=V$, a locally integrable potential growing like $|x|^{\alpha}$ for $0\leq \alpha<2$ at infinity. We establish that the Dyson-Phillips expansion converges in this case in an $r$ Schatten-von Neumann norm for $r>\frac{4}{2-\alpha}$ and show that $H_\vartheta+V$ is the generator of a Gibbs semigroup $\mathrm{e}^{-(H_\vartheta+V)\tau}$ for $|\arg{\tau}|\leq \frac{\pi}{2}-|\vartheta|$. From this we determine asymptotics for the eigenvalues and for the resolvent norm of $H_\vartheta+V$.