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Recently, Avron et al. in a series of papers shed new light on the question of quantum transport in mesoscopic samples coupled to particle reservoirs by semi-infinite leads. They rigorously treat the case, when the sample undergoes an adiabatic evolution thus generating a current through the leads, and prove the so-called BPT formula. Using a discrete model, we complement their work by giving a rigorous proof of the Landauer–Buttiker formula, which deals with the current generated by an adiabatic evolution on the leads. As is well known from physics, both of these formulas link the conductance coefficients for such systems to the S-matrix of the associated scattering problem. As an application, we discuss resonant transport through a quantum dot. The single charge tunneling processes are mediated by extended edge states, simultaneously localized near several leads.

A rigorous proof of the Landauer–Büttiker formula