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This document provides the mathematical details supporting the paper. \
Equation numbers and figures refer to the paper unless otherwise stated. The \
material is furnished with a description of the main steps but it goes \
without saying that the reader should flank this document with the paper in \
order to make profit out of this material.

The material is organized as follows. Sec. 1 develops the solution of the \
model in the scattering formalism, starting from the definition of the \
single-obstacle potential and the propagator matrix. Everything is worked out \
in terms of the abstract expressions for the propagators, that will be made \
explicit later. The analysis continues in Sec. 2 with the T-matrix in the \
plane wave basis. We begin our analysis by working out the solution of the \
non confined model in Secs. 3-5. We then proceed by considering the \
quasi-confined model. We commence with the propagators in Sec. 6 and we \
obtain the velocity response in Sec. 7 and the VACF in Sec. 8.

The quasi-confined model must reduce to the non-confined one when L->\
\[Infinity]. We also know that the quasi-confined model has less symmetry \
than the non-confined model; e.g., for the latter there is no distinction \
between the propagators g01 and g10, etc., whereas this distinction is \
essential in the quasi-confined model. Hence,  at the purely formal level, \
when we set g01=g10 (etc.) in a quantity specific for the quasi-confined \
model we should retrieve the non-confined model; this is tested in Sec. 9.

NB.
Please notice that the results of Sec. 4 contain, strictly speaking, also \
those for the non-confined model. The analysis of Sec. 4 is performed with \
the abstract propagators without assuming that g10=g01 (etc.), meaning that \
the results apply for the quasi-confined model. Results for the  non-confined \
model are obtained by restriction of the general solution by setting g10=g01 \
(etc.).\
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 "Equation (13) of PRL 2013 gives the terminal velocity in the linear \
response regime, therefore: v(t\[RightArrow]\[Infinity])=DF+corrections. Here \
we write the full analytical result and then we expand it for small forces, \
so we derive the corrections beyond linear response regime. It is convenient \
to write the terminal velocity as\nv(t\[RightArrow]\[Infinity]) = ",
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MSD response for the quasi-periodic model in terms of the abstract \
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