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
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Profiles

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