This thesis addresses the question of the time spent by a transmitted wave packet within a scattering region. The study involves mathematical aspects of solving the Schrodinger equation in open systems with a view to developing new conceptual approaches to scattering theory. Efficient schemes to obtain scattering matrices from mean-field Hamiltonians are developed and these are implemented in new numerical codes. The relationship between the phase of S-matrix elements and Wigner delay times is also elucidated.
I consider the scattering problem in a tight-binding lattice, as a simple way to understand the relation between M-functions and Greens functions and to investigate the connectivity dependence of Wigner delay times. To analyse delay times in bipartite lattices, tight binding calculations are used and a new computer code is developed to verify analytical predictions. In particular, Green’s functions and a mid-gap theory are used to calculate Wigner delay times for different connectivities in graphene like molecules. One interesting and counterintuitive result is that in the weak coupling limit at the middle of HOMO and LUMO gap, the Wigner delay time does not depend on the distance between the connections to external reservoirs.