Phase Flows Generated by Cauchy Problem for Nonlinear Schrödinger Equation and Dynamical Mappings of Quantum States
- Autores: Efremova L.1,2, Grekhneva A.3, Sakbaev V.2,1
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Afiliações:
- National Research Nizhny Novgorod State University
- Moscow Institute of Physics and Technology
- Gromov Flight Research Institute (GFRI)
- Edição: Volume 40, Nº 10 (2019)
- Páginas: 1455-1469
- Seção: Article
- URL: https://journals.rcsi.science/1995-0802/article/view/205641
- DOI: https://doi.org/10.1134/S1995080219100081
- ID: 205641
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Resumo
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.
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Sobre autores
L. Efremova
National Research Nizhny Novgorod State University; Moscow Institute of Physics and Technology
Autor responsável pela correspondência
Email: lefunn@gmail.com
Rússia, Nizhny Novgorod, 603950; Dolgoprudny, 141701
A. Grekhneva
Gromov Flight Research Institute (GFRI)
Autor responsável pela correspondência
Email: alice-prohorses@yandex.ru
Rússia, Zhukovskii, 140180
V. Sakbaev
Moscow Institute of Physics and Technology; National Research Nizhny Novgorod State University
Autor responsável pela correspondência
Email: fumi2003@mail.ru
Rússia, Dolgoprudny, 141701; Nizhny Novgorod, 603950