On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta

Abstract

We consider an inverse spectral problem for radial Schr\"odinger operators with singular potentials. First, we show that the knowledge of the Dirichlet spectra for infinitely many angular momenta~$\ell$ satisfying a M\"untz-type condition uniquely determines the potential. Next, in a neighborhood of the zero potential, we prove local uniqueness from two Dirichlet spectra associated with distinct angular momenta in the cases \((\ell_1,\ell_2) = (0,1)\,, \ (1,2)\) and \((0,3)\)\,. Our approach relies on an explicit analysis of the associated singular differential equation, combined with the classical Kneser--Sommerfeld formula. These results sharpen a theorem of Carlson-Shubin~(1994) and confirm, in the linearized setting and for these configurations, a conjecture originally formulated by Rundell and Sacks~(2001).