This is the first part of a series of papers dealing with eigenfunction expansions for the Schrödinger operator \(H=H_ 0+V(x)\), \(H_ 0=- \Delta /2\), where we assume V(x)\(\in {\mathcal B}^{\infty}(R^ n)\) is real-valued, \({\mathcal B}^{\infty}(R^ n)\) denoting the space of smooth functions on \(R^ n\) with bounded derivatives. In this part, we first construct the fundamental solution U(t,s) for the Schrödinger equation with time-dependent potential \(V(t)=V(t,x)\), V(t,x)\(\in {\mathcal B}^{\infty}(R^ n_ x)\) \((t\in R^ 1)\), \[ (D_ t+H(t))U(t,s)=0,\quad U(s,s)=I, \] where \(H(t)=H_ 0+V(t)\), in the form of Fourier integral operator \[ U(t,s)f(x)=Os-\int \int e^{i(x\xi -(t- s)\xi ^ 2/2-y\xi)} a(t,s,\xi,y)f(y) dy \partial \xi \] \[ =e^{i(t- s)H_ 0}\quad a(t,s,D_ x,X')f(x), \] for all \(t,s\in R^ 1\) and \(f\in {\mathcal S}\), where \(\partial \xi =(2\pi)^{-n}d\xi\). Then under the additional assumption that V(t,x) is real-valued, we prove the unitarity of \(a(t,s,D_ x,X')\) hence of U(t,s). These results will be used in Part II, and will be also useful for other investigations in Schrödinger theory.