2025/05/25 by Yuchao He, WU Meng-da, He, Yuchao +5 · 1 citation
Mathematics · Engineering · #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #Microwave Imaging and Scattering Analysis
paper · pdf · doi:10.48550/arxiv.2505.19129
This paper develops a methodological framework for addressing a novel and application-oriented inverse nodal problem in Sturm-Liouville operators, having significant applications in seismic wave analysis and submarine underwater radar (sonar) detection. By utilizing a given finite set of nodal data, we propose an optimization framework to find the potential q that is most closely approximating a predefined target potential q0. The inverse nodal optimization problem is reformulated as a solvability problem for a class of nonlinear Schrödinger equations, enabling systematic investigation of the inverse nodal problem. As an example, when the constant target potential q0 is considered, we find that the Schrödinger equations are completely integrable and conclude that the potential q is `periodic' in a certain sense. Furthermore, the reconstruction of q is reduced to solving a system of three featured parameters, thereby establishing an explicit quantitative relationship between ‖ q‖Lp and T_*. Of importance, we prove the uniqueness of the potential q when p>3/2. These new findings represent a substantial advancement in this field of study. Our methodology also bridges theoretical rigor with practical applicability, addressing scenarios where only partial nodal information is available.