subellipticity situations considered by Egorov and Hörmander, and the purpose of the present paper is to develop such an approach by studying an associated semi-group basically as a Fourier integral operator with complex phase in the spirit of Maslov [@Mas], Kucherenko [@Ku], Melin-Sjöstrand [@MeSj76]. (See also the more recent works by A. Menikoff-Sjöstrand [@MenSj], O. Matte [@Ma], extending the approach of [@MeSj76] to non-homogeneous cases.) Finally it turned out to be more convenient to use Bargmann-FBI transforms in the spirit of [@Sj82] and [@HeSj86]. The semigroup method led to a strengthened result in case (B): The resolvent can be extended to a disc of radius ${\cal O}((h\ln 1/h)^{k/(k+1)})$ around the appropriate boundary points. This improvement has been obtained recently by W. Bordeaux Montrieux [@Bo] for the model operator $hD_x+g(x)$, when $g\in C^\infty (S^1)$ and the points of maximum or minimum are all nondegenerate. In that case $k=2$ and Bordeaux Montrieux also constructed quasi-modes for values of the spectral parameter that are close to the boundary points.

We next state the results and outline the proof in case (B).

Let $X$ be equal to ${\bf R}^n$ or equal to a compact smooth manifold of dimension $n$.

In the first case, let 