  The main purpose of this paper is to show that the generalized Schr\"odinger
operator ${\cal A}^Vf={1/2}\Delta f+b\nabla f-Vf$, $f\in C_0^\infty(\R^d)$, is
a pre-generator for which we can prove its $L^\infty(\R^d,dx)$-uniqueness.
Moreover, we prove the $L^1(\R^d,dx)$-uniqueness of weak solutions for the
Fokker-Planck equation associated with this pre-generator.
