Entropic characterization of Tunneling and State Pairing in a Quasi-Exactly Solvable Sextic Potential

Abstract

We analyze the (de)localization properties of a quasi-exactly solvable (QES) sextic potential $V_{\text{QES}}(x)=\frac{1}{2}(x^6+2x^4-2(2\lambda +1)x^2)$ as a function of the tunable parameter $\lambda \in [-\frac{3}{4},6]$. For $\lambda >-\frac{1}{2}$, the potential exhibits a symmetric double-well structure, with tunneling emerging for the ground state level at $\lambda \approx 0.732953$. {For the lowest energy states \( n = 0,1,2,3 \), we construct physically meaningful variational wavefunctions that $i)$ respect parity symmetry under the transformation $x\to -x$, $ii)$ exhibit the correct asymptotic behavior at large distances, and $iii)$ allow for exact analytical Fourier transforms. Variational energies match Lagrange Mesh and available exact analytical QES results with relative errors $\simeq 10^{-8}$ for $n=0,1,2$ and $\simeq 10^{-6}$ for the third excited state $n=3$. We demonstrate that entropic measures (Shannon entropy, Kullback-Leibler, and Cumulative Residual Jeffreys divergences) surpass conventional variance-based methods in revealing tunneling transitions, wavefunction symmetry breaking, and quantum state pairing. Our results confirm that the Beckner-Bialynicki-Birula-Mycielski entropic uncertainty relation holds across all examined values of $n$ and $\lambda$. The quality of the trial function is also validated by the small $\sim 10^{-10}$ Cumulative Residual Jeffreys divergences from the exact QES solutions.

49 pages, 12 figures, 13 Tables