Optimization Inverse Spectral Problems for Sturm--Liouville Operators with Potentials in $L^p(0,1)$

We study an optimization inverse spectral problem (OISP) for a one-dimensional Sturm--Liouville operator on a finite interval. The problem is to find a potential $V \in L^p(0,1)$ that realizes a prescribed finite set $λ_1^* < λ_2^* < \dots < λ_m^*$ and has the smallest $L^p(0,1)$-distance from an a priori given potential $V_0 \in L^p(0,1)$. We formulate and prove results on the regularity of the isospectral manifold, the existence of an OISP solution, and an explicit analytic representation of the optimal potential by means of a power map applied to a linear combination of squared eigenfunctions. Using a unified differential-algebraic approach to squared eigenfunctions, we show that whenever these squares are linearly dependent, the corresponding potential can be expressed in terms of them and is an analytic function. For $p=2$, we prove that nondegenerate stationary points are isolated and that the number of global minima is finite provided that each of them is nondegenerate. For every $1