Phase Flows Generated by Cauchy Problem for Nonlinear Schrödinger Equation and Dynamical Mappings of Quantum States

Abstract

We consider the transformation of the initial data space for the Schrödinger equation. The transformation is generated by nonlinear Schrödinger 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 Schrödinger 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.