Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator

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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 −iϑx 2+e 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 |arg⁡τ|≤ [Formula presented] −|ϑ|≠ [Formula presented]. From this we determine high energy asymptotics for the eigenvalues and the resolvent norm of H ϑ+V.

Original languageEnglish
Article number108415
JournalJournal of Functional Analysis
Issue number7
Early online date27 Nov 2019
Publication statusPublished - 15 Apr 2020


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

ASJC Scopus subject areas

  • Analysis


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