English

Resolvent estimates for non-self-adjoint operators via semi-groups

Spectral Theory 2009-06-02 v1 Analysis of PDEs

Abstract

We consider a non-self-adjoint hh-pseudodifferential operator PP in the semi-classical limit (h0h\to 0). If pp is the leading symbol, then under suitable assumptions about the behaviour of pp at infinity, we know that the resolvent (zP)1(z-P)^{-1} is uniformly bounded for zz in any compact set not intersecting the closure of the range of pp. Under a subellipticity condition, we show that the resolvent extends locally inside the range up to a distance O(1)((hln1h)k/(k+1)){\cal O}(1)((h\ln \frac{1}{h})^{k/(k+1)}) from certain boundary points, where k{2,4,...}k\in \{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=2k=2. The method of proof is different from the one of Dencker et al, and is based on estimates of an associated semi-group.

Keywords

Cite

@article{arxiv.0906.0094,
  title  = {Resolvent estimates for non-self-adjoint operators via semi-groups},
  author = {Johannes Sjoestrand},
  journal= {arXiv preprint arXiv:0906.0094},
  year   = {2009}
}
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