2009/05/30 by Johannes Sjoestrand, Sjoestrand, Johannes
Mathematics · #35P05 #35P15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP) #math.AP #math.SP #msc:35P05 #msc:35P15
paper · pdf · doi:10.48550/arxiv.0906.0094
arxiv created 2009/05/30 · arxiv updated 2009/12/01
We consider a non-self-adjoint h-pseudodifferential operator P in the semi-classical limit (h→ 0). If p is the leading symbol, then under suitable assumptions about the behaviour of p at infinity, we know that the resolvent (z-P)-1 is uniformly bounded for z in any compact set not intersecting the closure of the range of p. Under a subellipticity condition, we show that the resolvent extends locally inside the range up to a distance \cal O(1)((hln (1)/(h))k/(k+1)) from certain boundary points, where k∈ \2,4,...\. This is a slight improvement of a result by Dencker, Zworski and the author, and it has recently been obtained by W. Bordeaux Montrieux in a model situation where k=2. The method of proof is different from the one of Dencker et al, and is based on estimates of an associated semi-group.