Spectrum of time operators

Let H be a self-adjoint operator on a complex Hilbert space H. A symmetric operator T on H is called a time operator of H if, for all t is an element of R, e(- i tH) D(T) subset of D(T) (D(T) denotes the domain of T) and Te- i tH psi = e- (i tH)(T + t) psi, for all t is an element of R, for all psi is an element of 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 is an element of C| Im z >= 0}. ( ii) If H is bounded above, then sigma(T) is either C or {z is an element of C| Im z <= 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.