Equilibrium configurations and stability analysis of a discrete toy model arising from the minimization of eigenvalues of the fractional Laplacian

We study a discrete toy problem introduced by Zahl in the context of minimizing eigenvalues of the fractional Laplacian. The model describes a system of points with signed masses interacting via the nonlocal potential |x|−(n+2s). We prove that the energy of any nondegenerate equilibrium must be zero and that an equilibrium necessarily contains both positive and negative masses. On this basis we establish two general theorems: first, every equilibrium is necessarily coplanar; second, for any coplanar equilibrium there exists a perturbation perpendicular to that plane which strictly decreases the energy. The construction uses a simple sign-function perturbation depending only on the coexistence of opposite signs. These results completely verify Zahl’s conjecture that this system has no local minima, and provide key insight into the global configuration of the original continuous problem.