Strong Time Operators Associated with Generalized Hamiltonians

Let the pair of operators, (H, T), satisfy the weak Weyl relation: $$T{\rm e}^{-itH}={\rm e}^{-itH}(T+t),$$ where H is self-adjoint and T is closed symmetric. Suppose that g is a real-valued Lebesgue measurable function on ${\mathbb {R}}$ such that ${g\in C^2(\mathbb {R}\backslash K)}$ for some closed subset ${K\subset\mathbb {R}}$ with Lebesgue measure zero. Then we can construct a closed symmetric operator D such that (g(H), D) also obeys the weak Weyl relation.