Commutators and self-adjointness of Hamiltonian operators

A time dependent approach to self-adjointness is presented and it is applied to quantum mechanical Hamiltonians which are not semi-bounded. Sufficient conditions are given for self-adjointness of Schrödinger and Dirac Hamiltonians with potentials which are unbounded at infinity. The method is the introduction of an auxiliary operatorN≧0 whose rate of change (commutator with the Hamiltonian) is bounded by a multiple ofN.