Estimation de r\'esolvante et construction de quasimode pr\`es du bord du pseudospectre

We consider a non-self-adjoint pseudodifferential operator in the semi-classical limit $(h\to 0)$. The principal symbol is given by p. We know that the resolvent $(z-P)^{-1}$ exists inside the range up to a distance $O((h\ln\frac{1}{h})^{\frac{k}{k+1}})$ from certain boundary points, where $k\in\{2,4,...\}$. In this work, we improve the resolvent estimates given by different authors in the case k=2 and in dimension one. For the proof, we will construct quasimodes by a scaling for z very close to the boundary. We give some applications of this formula.