  In this work we study the unitary time-evolutions of quantum systems defined
on infinite-dimensional separable time-dependent Hilbert spaces. Two possible
cases are considered: a quantum system defined on a stochastic interval and
another one defined on a Hilbert space with stochastic integration measure
(stochastic time-dependent scalar product). The formulations of the two
problems and a comparison with the general theory of open quantum systems are
discussed. Possible physical applications of the situations considered are
analyzed.
