2018/06/17 by Lyonell Boulton, Boulton, Lyonell
Computer Science · Engineering · Mathematics · #47D06 #81Q12 #81Q15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Spectral Theory (math.SP) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1806.06374
openalex publication_date 2018/06/17 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
Let T be the generator of a C0-semigroup e-Tt which is of finite\ntrace for all t>0 (a Gibbs semigroup). Let A be another closed operator,\nT-bounded with T-bound equal to zero. In general T+A might not be the\ngenerator of a Gibbs semigroup. In the first half of this paper we give\nsufficient conditions on A so that T+A is the generator of a Gibbs\nsemigroup. We determine these conditions in terms of the convergence of the\nDyson-Phillips expansion corresponding to the perturbed semigroup in suitable\nSchatten-von Neumann norms.\n In the second half of the paper we consider\nT=H_\ϑ=-e-i\ϑ\∂x2+ei\ϑx2, the\nnon-selfadjoint harmonic oscillator, on L2(\ℝ) and A=V, a locally\nintegrable potential growing like |x|\α for 0\≤ \α<2 at\ninfinity. We establish that the Dyson-Phillips expansion converges in this case\nin an r Schatten-von Neumann norm for r>\(4)/(2-\α) and show that\nH_\ϑ+V is the generator of a Gibbs semigroup\n\e-(H_\ϑ+V)\τ for |\arg\τ|\≤\n\(\π)/(2)-|\ϑ|. From this we determine asymptotics for the\neigenvalues and for the resolvent norm of H_\ϑ+V.\n