[en] We give a necessary and sufficient condition for the generalized Schroedinger operator to be essentially self-adjoint in L²(Ω; rhodx), under general assumptions on rho and for arbitrary domains Ω in Rsup(n). In particular, if rho is strictly positive and locally Lipschitz continuous on Ω = Rsup(n), then A is essentially self-adjoint. We also give examples of non-essential self-adjointness and a complete discussion of the one-dimensional case. These results have applications to the problem of the essential self-adjointness of quantum Hamiltonians and to the uniqueness problem of Markov processes. (orig./WL)