  It is shown, that the Aharonov-Albert-Vaidman concept of weak values appears
to be a consequence of a more general quantum phenomenon of weak quantum
evolution. Here the concept of weak quantum evolution is introduced and
discussed for the first time. In particular, it is shown on the level of
quantum evolution that there exist restrictions on the applicability of weak
quantum evolution- and, hence, weak values approach. These restrictions connect
the size of given quantum ensemble with the parameters of pre- and
post-selected quantum states. It is shown, that the latter requirement can be
fulfilled for the model system, where the concept of weak values was initially
introduced by Aharonov,Vaidman and Albert. Moreover, a deep connection between
weak quantum evolution and conventional probability of quantum transition
between two non-orthogonal quantum states is established for the first time. It
is found that weak quantum evolution of quantum system between its two
non-orthogonal quantum states is inherently present in the
measurement-determined definition of quantum transition probability between
these two quantum states.
