Time-Dependent Source Identification Problem for a Fractional Schrödinger Equationwith the Riemann–Liouville Derivative

Abstract

We consider a Schrödinger equation \(i{\partial }_{t}^{\rho }u\left(x,t\right)-{u}_{xx}\left(x,t\right)=p\left(t\right)q\left(x\right)+f\left(x,t\right),0<t\le T,0<\rho <1,\) with the Riemann–Liouville derivative. An inverse problem is investigated in which, parallel with u(x, t), a time-dependent factor p(t) of the source function is also unknown. To solve this inverse problem, we use an additional condition B[u(∙, t)] =ψ(t) with an arbitrary bounded linear functional B. The existence and uniqueness theorem for the solution to the problem under consideration is proved. The stability inequalities are obtained. The applied method makes it possible to study a similar problem by taking, instead of d²/dx², an arbitrary elliptic differential operator A(x,D) with compact inverse.