Identification of source terms in the Schr\"odinger equation with dynamic boundary conditions from final data

TL;DR

This paper addresses an inverse problem in the Schrödinger equation with dynamic boundary conditions, developing a theoretical framework for source term identification and validating it through numerical experiments.

Contribution

It introduces a weak solution approach and analyzes the Tikhonov functional to establish gradient formulas, existence, and uniqueness for the inverse source problem.

Findings

Proved Lipschitz continuity of the gradient of the Tikhonov functional.

Established existence and uniqueness of a quasi-solution.

Validated the approach with numerical experiments in one dimension.

Abstract

In this paper, we study an inverse problem of identifying two spatial-temporal source terms in the Schr\"odinger equation with dynamic boundary conditions from the final time overdetermination. We adopt a weak solution approach to solve the inverse source problem. By analyzing the associated Tikhonov functional, we prove a gradient formula of the functional in terms of the solution to a suitable adjoint system, allowing us to obtain the Lipschitz continuity of the gradient. Next, the existence and uniqueness of a quasi-solution are also investigated. Finally, our theoretical results are validated by numerical experiments in one dimension using the Landweber iteration method.