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

TL;DR

This paper addresses an inverse problem for a fractional Schrödinger equation with Riemann-Liouville derivatives, establishing existence, uniqueness, and stability of solutions, and extending the method to general elliptic operators.

Contribution

It introduces a novel approach to identify a time-dependent source in a fractional Schrödinger equation with proven theoretical guarantees.

Findings

Proved existence and uniqueness of the inverse problem solution.

Derived stability inequalities for the solution.

Extended the method to general elliptic operators.

Abstract

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.