### Abstract

Let T be the generator of a C
_{0}-semigroup e
^{−Tt} which is of trace class 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 in suitable Schatten-von Neumann norms. In the second half of the paper we consider T=H
_{ϑ}=−e
^{−iϑ}∂
_{x}
^{2}+e
^{iϑ}x
^{2}, the non-selfadjoint harmonic oscillator, on L
^{2}(R) and A=V, a locally integrable potential growing like |x|
^{α} at infinity for 0≤α<2. We establish that the Dyson-Phillips expansion converges in r Schatten-von Neumann norm in this case for r large enough and show that H
_{ϑ}+V is the generator of a Gibbs semigroup e
^{−(H
ϑ+V)τ
} for |argτ|≤ [Formula presented] −|ϑ|≠ [Formula presented]. From this we determine high energy asymptotics for the eigenvalues and the resolvent norm of H
_{ϑ}+V.

Original language | English |
---|---|

Article number | 108415 |

Journal | Journal of Functional Analysis |

Early online date | 27 Nov 2019 |

DOIs | |

Publication status | E-pub ahead of print - 27 Nov 2019 |

### Keywords

- Dyson-Phillips expansion
- Non-selfadjoint Schrödinger operators
- Perturbation of Gibbs semigroups

### ASJC Scopus subject areas

- Analysis

## Fingerprint Dive into the research topics of 'Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator'. Together they form a unique fingerprint.

## Profiles

## Lyonell Boulton

- School of Mathematical & Computer Sciences - Associate Professor
- School of Mathematical & Computer Sciences, Mathematics - Associate Professor

Person: Academic (Research & Teaching)