Finite-data inverse nodal optimization in angular-momentum sectors of Schrödinger operators

Abstract

In this paper, we study a finite-data inverse nodal optimization problem for radial Schrödinger operators $$ H_q:=-Δ+q(|x|),\qquad u|_{\partial B_R}=0, \qquad d\geq2, $$ on the ball $B_R\subset\mathbb R^d$, in an arbitrary fixed angular-momentum sector. The analysis is built directly at the Friedrichs endpoint and in the physical weighted space $L_d^p$, $p>d/2$, so that the singular radial geometry is retained rather than replaced by a regular one-dimensional model. By means of a Volterra representation of the Friedrichs branch, we establish weak continuity and continuous Fréchet differentiability of fixed nodal radii with respect to the potential. We then prove existence of minimum-distance potentials and derive the corresponding critical Schrödinger equations. For several nodes of the same eigenfunction, the nodal gradients are linearly independent, which yields a finite-dimensional submersion structure and sharp local minimum-norm reconstruction. A general observation principle is further used to treat mixed angular-momentum data under the corresponding transversality condition and simultaneous spectral--nodal data from one eigenmode, for which transversality is automatic; in the Hilbert case this gives explicit inverse-Gram formulas and local uniqueness. Finally, when $\ell=0$, the reference potential is constant, and the second radial mode is considered, we prove that every inward displacement of the unique interior node admits a unique global optimizer for $p>(d+2)/2$.