Half-inverse problems for perturbed Bessel operators with a known left-hand potential

For the regular Sturm-Liouville operator - d² dx²+q acting in L²(0,1), the Hochstadt-Lieberman theorem states that one full spectrum together with the potential on either the left half-interval q|(0,1/2) or the right half-interval q|(1/2,1) uniquely determines the full potential q. The inverse-square singular term l(l+1)/x² in the perturbed Bessel operator - d² dx²+l(l+1)/x²+q, acting in L²(0,1), breaks the left-right symmetry. It is known that, when the right half of the potential q|(1/2,1) is given, the uniqueness of the potential can be established even if a finite number of eigenvalues (depending on l) are omitted. In this paper we study the opposite setting--where the left half potential q|(0,1/2) is known--and show that one full spectrum together with finitely many norming constants uniquely determine the full potential q. Moreover, we specify the minimum number of norming constants required for the uniqueness. Finally, for positive integer l, L² potentials, and the Dirichlet boundary condition at $x = 1$, we establish the local bi-Lipschitz stability and the local solvability from the full spectrum and finitely many norming constants, assuming the potential on the left half-interval is fixed.