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arXiv · 2509.26485

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

Damien Gobin·Benoît Grébert·Bernard Helffer·François Nicoleau
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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).

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Damien Gobin, Benoît Grébert, Bernard Helffer, François Nicoleau. 2025-09-30. On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta. https://arxiv.org/abs/2509.26485

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