uniformly bounded for $h>0$ small enough and for $z$ in any fixed compact set in ${\bf C}$, disjoint from the closure of the range of $p$. On the other hand, by a quasi-mode construction of E.B. Davies [@Da], that was generalized by Zworski [@Zw] by reduction to an old quasi-mode construction of Hörmander (see also [@DeSjZw] for a more direct approach), we also know that if ${\bf C}\ni z=p(\rho )$, where $\rho $ is a point in phase space where $i^{-1}\{p,\overline{p}\}>0$ and $\{\cdot ,\cdot \cdot \}$ denotes the Poisson bracket, then we have quasimodes for $P-z$ in the sense that there exist $u=u_h\in C_0^\infty $, normalized in $L^2$, such that the $L^2$ norm of $(P-z)u_h$ is ${\cal O}(h^\infty )$, implying, somewhat roughly, that the norm of the resolvent (whenever it is defined) cannot be bounded by a negative power of $h$.

A natural question is then what happens when $z$ is close to the boundary of the range of $p$. L. Boulton [@Bo] and Davies [@Da2] obtained some results about this in the case of the non-self-adjoint harmonic operator on the real line. As with the quasi-mode construction this question is closely related to classical results in the general theory 