Multidimensional inverse nodal theory for Schrödinger operators

Develop a multidimensional inverse nodal theory for Schrödinger operators, extending beyond the radial setting in balls and addressing the recovery of multidimensional potentials from eigenfunction nodal-set data.

Background

The paper studies a radial specialization of an inverse nodal problem for Schrödinger operators on balls. Under radial symmetry, nodal hypersurfaces are concentric spheres, and their radii can be analyzed through an exact reduction to a singular weighted Sturm–Liouville problem.

The authors identify a general multidimensional theory as unresolved and treat the radial case as a tractable step toward that broader objective. The open problem concerns formulating and developing inverse nodal methods for genuinely multidimensional Schrödinger operators, where nodal sets have nontrivial hypersurface geometry rather than being described by an ordered sequence of radii.