Asao Arai
Spectrum of Time Operators
(28K, Latex2e)
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.