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) –$\mathrm{\Delta }$ + U (V $\ge$ 0) and –$\mathrm{\Delta }$ + iV in L 2 (R^d) for C $\mathrm{\infty }$ 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, $\mathrm{\forall }$u $\in$ C^$\mathrm{\infty }$_0 (R ^d), ||u||^2 _{H^2 (R^d)} + ||U u||^2 _{L^2 (R^d)}$\le$ C ||(–$\mathrm{\Delta }$ + U)u||^2_{L^2 (R^d)} + ||u||^2_{L^2 (R^d)}or similar estimates for $-\mathrm{\Delta }+iV$.