The boundary value problem \[ u(x, t)_t- i\sum (a_{ij}(x, t) u(x,t)_{x_i})_{x_j}- c(x,t) u(x,t)= f(x,t)\;\text{in }Q, \] \[ u(x,t)= 0\;\text{on }\Sigma,\quad u(x,0)= u_0(x)\;\text{on }Q_0, \] is investigated, where \(Q\subset \mathbb{R}^n\times (0,T)\) is an open subset (with the lateral boundary denoted by \(\Sigma\)) whose sections \(Q_t\) (\(t\) fixed) satisfy the nondecreasing property \(Q_s\subset Q_t\) \((s0} \text{int }Q_t\) a nonempty set. Moreover, \(a_{ij}\in L^\infty(Q)\), \(c\in L^\infty(Q, \mathbb{R})\), \(a_{ij}=\overline a_{ji}\), and the uniform ellipticity \(\sum a_{ij}\xi_i\overline\xi_j> a|\xi|^2\) are supposed. Result: If \(u_0\in L^2(Q_0)\) and \(f\in L^2(Q)\), then the problem admits at most one solution satisfying the conservation law \[ \int_{Q_T}|u(x, T)|^2 dx= \int_{Q_0}|u_0(x)|^2 dx+ 2\text{Re} \int_Q f(x,t)\overline{u(x, t)} dx dt. \] If futhermore \(u_0\in H^1_0(Q_0)\) and \(f_t\in L^2(Q)\), then such a solution exists. The result is derived by using an abstract variational theory of the Schrödinger-type equation in Hilbert spaces.