Identification of source terms in the Schrödinger equation with dynamic boundary conditions from final data

Abstract

In this paper, we study an inverse problem of identifying two spatial-temporal source terms in the Schrödinger 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.

Figures (6)

• Figure 1: Real part (left) and imaginary part (right) of the solution $Y(x,t;f)$ to \ref{['1deq1to4']}.
• Figure 2: Exact and recovered $\operatorname{Re} f(x)$ (left) and $\operatorname{Im} f(x)$ (right) for $p\in \{1\%,3\%,5\%\}$.
• Figure 3: Real part (left) and imaginary part (right) of the solution $Y(x,t;f)$ to \ref{['1deq1to4']}.
• Figure 4: Exact and recovered $\operatorname{Re} f(x)$ (left) and $\operatorname{Im} f(x)$ (right) for $p\in \{1\%,3\%,5\%\}$.
• Figure 5: Real part (left) and imaginary part (right) of the solution $Y(x,t;f)$ to \ref{['1deq1to4']}.

Theorems & Definitions (21)

• Proposition 1.1
• Proposition 1.2
• proof
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Corollary 3.1
• Lemma 3.2
• Remark 4.1