  We investigate the relaxation dynamics in an isolated quantum many-body
system. The stationary value of an observable after relaxation is a topic of
researches in the field of quantum thermalization, since thermalization is a
relaxation phenomena where this stationary value coincides with the equilibrium
value. Therefore, computing the stationary value in quantum many-body systems
is regarded as an important problem. We, however, prove that the stationary
value in quantum many-body systems is incomputable. More precisely, we show
that whether the stationary value is in the vicinity of a given value or not is
an undecidable problem. Our undecidable result is still satisfied when we
restrict our system to a one-dimensional shift-invariant system with
nearest-neighbor interaction, our initial state to a product state of a state
on a single site, and our observable to a shift-sum of a one-body observable.
This result clearly shows that there is no general theorem or procedure to
decide the presence or absence of thermalization in a given quantum many-body
system.
