A young scholar from Donghua University’s School of Mathematics and Statistics has reported new progress in the spectral theory of ordinary differential equations (ODEs), contributing fresh insights to a core area of mathematical analysis.
Zhang Zhi, working with collaborators, has published a paper in the leading international journal Advances in Mathematics titled “Extremal Norms of Potentials from Fixed Eigenvalues or Eigenvalue Ratios for Camassa–Holm Equations.” The journal is widely regarded as a premier venue for significant advances in pure mathematics.
The study addresses an inverse spectral problem for a class of second-order ODEs, focusing on how to optimize extremal Lebesgue norms of potential functions under constraints involving fixed eigenvalues or eigenvalue ratios. The problem is closely connected to the spectral analysis of the Lax pair associated with the Camassa–Holm equation, a well-known model describing shallow water wave dynamics.
Building on the theory of measure differential equations, the research team developed a novel analytical framework that removes conventional restrictions such as symmetry or convexity assumptions on potential functions. This broader approach significantly expands the applicability of the results. Peer reviewers commended the work, noting that “the technical level is first-class and the results are of interest to several active research groups.”
The paper extends Zhang’s earlier contribution, also published in Advances in Mathematics, “Minimizations of Positive Periodic and Dirichlet Eigenvalues for General Indefinite Sturm–Liouville Problems,” which laid the groundwork for the present findings.
The latest results further reinforce Donghua University’s growing strength in spectral theory research and open new avenues for future exploration in the field.
