Abstract: The time operator is informally defined as a symmetric operator satisfying CCR with a given self-adjoint operator H, i.e. [H,T]=i[H,T]=i. However it is nontrivial to construct T associated with H, and W. Pauli mentioned at 1933 that there was no quantum time operators associated with H having eigenvalues or being semi-bounded. In this talk we define 5 classes of time operators from mathematical point of view, and show that a time operator exists for a Schroedinger operator H=−Δ+VH=−Δ+V with some VV. Note that H is semi-bounded and has infinitely many eigenvalues.