Sojourn-time operator approach to interaction time in quantum scattering: General formulation for arbitrary scattering systems

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Abstract
A general formulation of the sojourn-time operator approach is presented, valid for arbitrary scattering systems and free of restrictions to one-dimensional systems present in the previous work of the authors. We show how first-order perturbation theory can be consistently formulated in terms of appropriate sojourn-time operators. Both time-dependent and time-independent perturbations, including coupling between translation and internal degrees of freedom, are considered. The mean values of these Hermitian sojourn-time operators give the mean sojourn time (dwell time) of the particle in spatial regions or, more generally, in certain subspaces of the state space of the system defined by sets of commuting observables. The perturbed S operator and also the effect of the perturbation on the change of observables due to the scattering are, to a first-order approximation, fully expressible in terms of these operators. We also show that this approach provides a unified treatment of the various idealized one-dimensional systems, which have frequently been employed in the debate on tunneling and interaction times. In particular, we provide some insight into the role of complex times, which appear as combinations of matrix elements of the sojourn-time operators.