We study the (de)localization properties of a quasi-exactly solvable (QES) sextic potential \(V_{\text {QES}}(x) = \tfrac{1}{2}(x^6 + 2x^4 - 2(2\lambda + 1)x^2)\) as a function of the tunable parameter \(\lambda \in [-\tfrac{3}{4},6]\) . For the lowest states \(n=0,1,2,3\) , variational wavefunctions are constructed that respect parity, have correct asymptotics, and admit analytical Fourier transforms. Energies agree with Lagrange-mesh and exact QES results with relative errors of order \(10^{-8}\) for \(n=0,1,2\) and \(10^{-6}\) for \(n=3\) . We show that information-theoretic indicators (Shannon entropy, Kullback–Leibler and Cumulative Residual Jeffreys divergences) are more sensitive than variance-based measures in detecting tunneling transitions, symmetry breaking, and level pairing. The Beckner–Białynicki-Birula–Mycielski entropic uncertainty relation is verified across all \(\lambda\) , and trial densities are validated by divergences as small as \(10^{-10}\) from exact solutions. Importantly, while QES solvability allows for analytic benchmarking, many of the informational signatures identified here—such as entropy sensitivity and divergence-based detection of quasi-degeneracy—are generic to symmetric double-well systems and not restricted to the QES framework.
