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, Tbounded with Tbound 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 DysonPhillips expansion in suitable Schattenvon Neumann norms. In the second half of the paper we consider T=H _{ϑ}=−e ^{−iϑ}∂ _{x} ^{2}+e ^{iϑ}x ^{2}, the nonselfadjoint 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 DysonPhillips expansion converges in r Schattenvon 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 
Volume  278 
Issue number  7 
Early online date  27 Nov 2019 
DOIs  
Publication status  Published  15 Apr 2020 
Keywords
 DysonPhillips expansion
 Nonselfadjoint Schrödinger operators
 Perturbation of Gibbs semigroups
ASJC Scopus subject areas
 Analysis
Fingerprint
Dive into the research topics of 'Perturbations of Gibbs semigroups and the nonselfadjoint harmonic oscillator'. Together they form a unique fingerprint.Profiles

Lyonell Boulton
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)