On the Existence of the Dynamics for Anharmonic Quantum Oscillator Systems

(1.1) i∂tψ(t) = Hψ(t) . For all initial conditions ψ(0) ∈ H, the unique solution is given by ψ(t) = e−itHψ(0), for all t ∈ R. Due to Stone’s Theorem e−itH is a strongly continuous one-parameter group of unitary operators on H, and the self-adjointness of H is the necessary and sufficient condition for the existence of a unique continuous solution for all times. An alternative description of this dynamics is the so-called Heisenberg picture in which the time evolution is defined on the algebra of observables instead of the Hilbert space of states. The corresponding Heisenberg equation is