H.D. Cornean, A. Jensen, V. Moldoveanu
A rigorous proof for the Landauer-B\"uttiker formula
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ABSTRACT. Recently,
Avron {\it et al} in \cite{Avron1},...,\cite{Avron5} 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, see \cite{BPT}.
Using a discrete model, we complement their work by giving a
rigorous proof of the
Landauer-B\"uttiker formula, which deals with the current generated by
an adiabatic evolution on the leads. As it is well known in physics,
these
formulae link the conductance coefficients for such systems to
the $S$-matrix of the associated scattering problem.
As an application, we discuss the resonant transport through a
quantum dot.
The single charge tunneling processes are mediated by extended edge
states simultaneosly localized near several leads.