On the domain of a magnetic Schrödinger operator with complex electric potential

The aim of this paper is to review and compare the spectral properties of (the closed extension of) −∆ + U (V ≥ 0) and −∆ + iV in L 2 (R^d) for C ∞ 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: • essential selfadjointness or maximal accretivity • Compactness of the resolvent. • Maximal inequalities, i.e. the existence of C > 0 such that, ∀u ∈ C^∞_0 (R ^d), ||u||^2 _{H^2 (R^d) } + ||U u||^2 _{L^2 (R^d) }≤ C ||(−∆ + U)u||^2_{ L^2 (R^d)} + ||u||^2_{ L^2 (R^d) }or similar estimates for $-\Delta + i V$