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

• URL: http://arxiv.org/abs/2506.22684v2
• Date: Tue, 08 Jul 2025 04:39:32 GMT
• Title: Entropic characterization of Tunneling and State Pairing in a Quasi-Exactly Solvable Sextic Potential
• Authors: Angelina N. Mendoza Tavera, Adrian M. Escobar Ruiz, Robin P. Sagar,
• Abstract summary: We analyze a quasi-exactly solvable (QES) sextic potential $V_textQES(x) = frac12(x6 + 2x4 - 2(2lambda + 1)x2)$ as a function of the parameter $lambda in [-frac34, 6]$.<n>For $lambda > -frac12$, the potential exhibits a symmetric double-well structure, with tunneling emerging for the ground state level at $lambda approx
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• License: http://creativecommons.org/licenses/by/4.0/
• 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 $x \rightarrow -x$, $ii)$ 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 $\sim10^{-10}$ Cumulative Residual Jeffreys divergences from the exact QES solutions.

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