Time-dependent source identification problem for a fractional Schrodinger equation with the Riemann-Liouville derivative

Abstract

The Schrödinger equation $i{\partial }_t^{\rho }u(x,t)-u_{xx}(x,t)=p(t)q(x)+f(x,t)$ ( $0<t\le T,0<\rho <1$), 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 $d^2/d{x}^2$ an arbitrary elliptic differential operator $A(x,D)$, having a compact inverse.

Schrodinger type equations