  The Schr\"odinger equation $i \partial_t^\rho u(x,t)-u_{xx}(x,t) = p(t)q(x) +
f(x,t)$ ( $0<t\leq T, \, 0<\rho<1$), with the Riemann-Liouville derivative is
considered. An inverse problem is investigated in which, along with $u(x,t)$,
also a time-dependent factor $p(t)$ of the source function is unknown. To solve
this inverse problem, we take the additional condition $ B [u (\cdot,t)] = \psi
(t) $ with an arbitrary bounded linear functional $ B $. Existence and
uniqueness theorem for the solution to the problem under consideration is
proved. Inequalities of stability are obtained. The applied method allows us to
study a similar problem by taking instead of $d^2/dx^2$ an arbitrary elliptic
differential operator $A(x, D)$, having a compact inverse.
