Self-adjoint approximations of the degenerate Schrödinger operator
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- 作者：V. Zh. Sakbaev ; I. V. Volovich
- 关键词：Key wordsthe degenerate Schrödinger operator ; C* ; algebra ; pure and mixed states
- 刊名：P-Adic Numbers, Ultrametric Analysis, and Applications
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：9
- 期：1
- 页码：39-52
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Algebra;
- 出版者：Pleiades Publishing
- ISSN：2070-0474
- 卷排序：9
文摘
The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.