  Recently, the notions of Weak Measurement (WM), Weak Value (WV) and
Two-State-Vector Formalism (TSVF), firstly introduced by Aharonov and
collaborators, have extended the theoretical frame of standard quantum
mechanics, thus providing a quantum-theoretical formalism for extracting new
information from a system in the limit of small disturbance to its state. Here
we provide an application to the case of two-body scattering with one body
weakly interacting with its environment --- e.g. a neutron being scattered from
a H2 molecule physisorbed in a carbon nanotube.
  In particular, we make contact with the field of incoherent inelastic neutron
scattering from condensed systems. We provide a physically compelling
prediction of a new quantum effect --- a momentum transfer deficit; or
equivalently, an enhanced energy transfer; or an apparent reduction of the mass
of the struck particle. E.g., when a neutron collides with a H2 molecule in a
C-nanotube and excites its translational motion along the nanotube, it
apparently exchanges energy and momentum with a fictitious particle with mass
of 0.64 atomic mass units. Experimental results are shown and discussed in the
new theoretical frame. The effect under consideration has no conventional
interpretation, thus also supporting the novelty of the quantum theoretical
framework of WV and TVSF. Some speculative remarks about possible applications
being of technological interest (fuel cells and hydrogen storage; Li+
batteries; information and communication technology) are shortly mentioned.
\end{abstract}
