Hokkaido University | Library | HUSCAP Advanced Search 言語 日本語 English

# Generalized Weak Weyl Relation and Decay of Quantum Dynamics

Files in This Item:
 pre715.pdf 210.5 kB PDF View/Open
 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 time-energy uncertainty relation Schroedinger operator Dirac operator Fock space second quantiation. Issue Date: 12-Apr-2005 Journal Title: Hokkaido University Preprint Series in Mathematics Volume: 715 Start Page: 1 End Page: 37 Abstract: Let $H$ be a self-adjoint operator on a Hilbert space ${\cal H}$, $T$ be a symmetric operator on ${\cal H}$ and $K(t)$ ($t\in \R$) be a bounded self-adjoint 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{time-energy 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