# Time-dependent source identification problem for a fractional Schrödinger equation with the Riemann–Liouville derivative

### Abstract

UDC 517.9

We consider a Schrödinger 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. 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 (\cdot,t)] = \psi (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 make it possible to study a similar problem by taking, instead of $d^2/dx^2,$ an arbitrary elliptic differential operator $A(x, D)$ with compact inverse.

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