We consider the transformation of the initial data space for the Schrodinger equation. The transformation is generated by nonlinear Schrodinger operator on the segment [−π, π] satisfying the homogeneous Dirichlet conditions on the boundary of the segment. The potential here has the type $\xi(u)=\left(1+|u|^{2}\right)^{\frac{p}{2}} u$ , where u is an unknown function, p ≥ 0. The Schrodinger operator defined on the Sobolev space $$H_0^2([-\pi, \pi])$$ generates a vector field $${\rm{v}}:H_0^2([-\pi, \pi])\rightarrow{H}\equiv{L_2}(-\pi, \pi)$$ . First, we study the phenomenon of global existence of a solution of the Cauchy problem for p ∈ [0, 4) and, second, the phenomenon of rise of a solution gradient blow up during a finite time for p ∈ [4, +∞). In second case we study qualitative properties of a solution when it approaches to the boundary of its interval of existence. Moreover, we define a solution extension through the moment of a gradient blow up using both the one-parameter family of multifunctions and the one-parameter family of probability measures on the initial data space of the Cauchy problem. We show that this extension describes the destruction of a solution as the destruction of a pure quantum state and the transition from the set of pure quantum states into the set of mixed quantum states.