Spectral Theorem Approach to the Characteristic Function of Quantum Observables

We compute the resolvent of anti-commutator operator $XP+PX$ and quantum harmonic oscillator Hamiltonian $\frac{1}{2}(X^2+P^2)$. Using Stone's formula for finding spectral resolution an, either bounded or unbounded, self-adjoint on a Hilbert space, we also their Vacuum Characteristic Function (Quantum Fourier Transform). show how is applied to computation finite dimensional observables. The method proposed as an analytical alternative algebraic (or Heisenberg) approach relying associated Lie algebra commutation relations.