Spectrum of Time Operators

Abstract

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