Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation

Abstract

We study an inverse problem related to the dynamical Schr{ö}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schrödinger equation possesses a trivial solution, we linearize the equation around the trivial solution. Demonstrating the well-posedness of the direct problem under appropriate conditions on initial and boundary data, it is observed that the solution admits $\eps$-expansion. By taking into account the fact that the terms $\Oh(|\nabla u(t,x)|^3)$ are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vector-valued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion.

There is an issue with the proof of one of the Theorem in the paper, because of which result is incomplete