Selfadjoint time operators and invariant subspaces

Abstract

For classical dynamical systems time operators are introduced as selfadjoint operators satisfying the so called weak Weyl relation with the unitary groups of time evolution. Dynamical systems with time operators are intrinsically irreversible because they admit Lyapounov operators as functions of the time operator. For quantum systems selfadjoint time operators are defined in the same way or as dilations of symmetric ones dealing with the time-energy uncertainty relation, times of occurrence or survival probabilities. This work tackles the question of the existence of selfadjoint time operators on the basis of the Halmos-Helson theory of invariant subspaces, the Sz.-Nagy-Foias dilation theory and the Misra-Prigogine-Courbage theory of irreversibility. It is shown that the existence of a selfadjoint time operator for a unitary evolution is equivalent to the intrinsic irreversibility of the evolution plus the existence of a simply invariant subspace or a rigid operator-valued function for its Sz.-Nagy-Foias functional model. An extensive set of equivalent conditions to the existence of selfadjoint time operators can be obtained from these results. Such conditions are written in terms of Schrödinger couples, the Weyl commutation relation, incoming and outgoing subspaces, innovation processes, Lax-Phillips scattering processes, and translation and spectral representations. As an example, the selfadjoint extension of the quantum Aharonov-Bohm time-of-arrival operator is studied.