We present the theory of positive commutator and its recent improvements. We discuss applications to the spectral analysis of magnetic Laplacians on manifolds, singular Dirac operators, and slowly decaying Schroedinger operators. We also study the question of various discrete Laplacians operators for the question of the essential self-adjointness and the asymptotic of eigenvalues. Then we present some results related to the question of the absolutely continuous spectrum for discrete 1-dimensional Dirac operator. Finally we give a characterisation of Hamiltonian path for higher dimensionnal chessboards.
