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Generalized Weak Weyl Relation and Decay of Quantum Dynamics
Title:  Generalized Weak Weyl Relation and Decay of Quantum Dynamics 
Authors:  Arai, Asao Browse this author 
Keywords:  generalized weak Weyl relation  time operator  canonical commutation relation  Hamiltonian  quantum dynamics  survival probability  decay in time  timeenergy uncertainty relation  Schroedinger operator  Dirac operator  Fock space  second quantiation. 
Issue Date:  12Apr2005 
Journal Title:  Hokkaido University Preprint Series in Mathematics 
Volume:  715 
Start Page:  1 
End Page:  37 
Abstract:  Let $H$ be a selfadjoint operator on a Hilbert space ${\cal H}$, $T$ be a symmetric operator on ${\cal H}$ and $K(t)$ ($t\in \R$) be a bounded selfadjoint operator on ${\cal H}$. We say that $(T,H,K)$ obeys the {\it generalized weak Weyl relation} (GWWR) if $e^{itH}D(T) \subset D(T)$ for all $t \in \R$ and $Te^{itH}\psi=e^{itH}(T+K(t))\psi, \forall \psi \in D(T)$ ( $D(T)$ denotes the domain of $T$). In the context of quantum mechanics where $H$ is the Hamiltonian of a quantum system, we call $T$ a {\it generalized time opeartor} of $H$. We first investigate, in an abstract framework, mathematical structures and properties of triples $(T,H,K)$ obeying the GWWR. These include the absolute continuity of the spectrum of $H$ restricted to a closed subspace of ${\cal H}$, an uncertainty relation between $H$ and $T$ (a \lq\lq{timeenergy uncertainty relation}"), the decay property of transition probabilities $\left\lang \psi,e^{itH}\phi\rang \right^2$ as $t \to \infty$ for all vectors $\psi$ and $\phi$ in a subspace of ${\cal H}$. We describe methods to construct various examples of triples $(T,H,K)$ obeying the GWWR. In particular we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces, fermion Fock spaces) which may have applications to quantum field models with interactions. 
Type:  bulletin (article) 
URI:  http://hdl.handle.net/2115/69520 
Appears in Collections:  理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

Submitter: 数学紀要登録作業用
