Invariant Domains for the Time - Dependent Schrijdinger Equation

have global solutions. Results of Saari [9] and Sperling [II] imply, however, that the basic problem in proving this classical result would be the proof that, for almost every initial condition, x(t)” = XL, xj(t>” cannot become infinite in finite time. Interestingly enough, while Kato’s result “solves” the dynamical existence question in the quantum case, it says nothing about the question of x(t)” remaining finite in time! From its physical interpretation, proof of such a regularity property is clearly desirable. The problem to which we address ourselves here is, therefore, that of finding a dense set of physically reasonable states, f, such that (exp(-iHi)f, xz exp(-iiNt)f:> remains finite in finite time. The solution of this problem is not