07-26 Asao Arai

Spectrum of Time Operators (28K, Latex2e) Jan 31, 07

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Abstract. Let $H$ be a self-adjoint operator on a complex Hilbert space ${\cal H}$. A symmetric operator $T$ on ${\cal H}$ is called a time operator of $H$ if, for all $t\in \R$, $e^{-itH}D(T)\subset D(T)$ ($D(T)$ denotes the domain of $T$) and $Te^{-itH}\psi=e^{-itH}(T+t)\psi, \ \forall t\in \R, \forall \psi \in D(T)$. In this paper, spectral properties of $T$ are investigated. The following results are obtained: (i) If $H$ is bounded below, then $\sigma(T)$, the spectrum of $T$, is either $\C$ (the set of complex numbers) or $\{z\in \C| \Im z \geq 0\}$. (ii) If $H$ is bounded above, then $\sigma(T)$ is either $\C$ or $\{z\in \C| \Im z \leq 0\}$. (iii) If $H$ is bounded, then $\sigma(T)=\C$. The spectrum of time operators of free Hamiltonians for both nonrelativistic and relativistic particles is exactly identified. Moreover spectral analysis is made on a generalized time operator.

Files: 07-26.src( 07-26.keywords , spectrum.tex )