On the domain of a magnetic Schr\"odinger operator with complex electric potential

The aim of this paper is to review and compare the spectral properties of (the closed extension of) --$\Delta$ + U (V $\ge$ 0) and --$\Delta$ + iV in L 2 (R^d) for C $\infty$ real potentials U or V with polynomial behavior. The case with magnetic field will be also considered. More precisely, we would like to present the existing criteria for: $\bullet$ essential selfadjointness or maximal accretivity $\bullet$ Compactness of the resolvent. $\bullet$ Maximal inequalities, i.e. the existence of C > 0 such that, $\forall$u $\in$ C^$\infty$\_0 (R ^d), ||u||^2 \_{H^2 (R^d)} + ||U u||^2 \_{L^2 (R^d)}$\le$ C ||(--$\Delta$ + U)u||^2\_{L^2 (R^d)} + ||u||^2\_{L^2 (R^d)}or similar estimates for $-\Delta + i V$.