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arXiv:2506.22684 · International Journal of Theoretical Physics

Entropic Characterization of Tunneling and State Pairing in a Quasi-exactly Solvable Sextic Potential

Angelina N. Mendoza Tavera, A. M. Escobar-Ruiz, Robin P. Sagar

Published Nov 17, 2025• 2 citations
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Abstract

Abstract 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.

DOI
10.1007/s10773-025-06189-x

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Abstract We study the (de)localization properties of a quasi-exactly solvable (QES) sextic potential VQES(x)=12(x6+2x4−2(2λ+1)x2)V_{\text {QES}}(x) = \tfrac{1}{2}(x^6 + 2x^4 - 2(2\lambda + 1)x^2)VQES​(x)=21​(x6+2x4−2(2λ+1)x2) as a function of the tunable parameter λ∈[−34,6]\lambda \in [-\tfrac{3}{4},6]λ∈[−43​,6] . For the lowest states n=0,1,2,3n=0,1,2,3n=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 10−810^{-8}10−8 for n=0,1,2n=0,1,2n=0,1,2 and 10−610^{-6}10−6 for n=3n=3n=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 λ\lambdaλ , and trial densities are validated by divergences as small as 10−1010^{-10}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.
Abstract We develop a phase-space description of tunneling in the quasi-exact sextic double well VQES(x)=12 (x6+2x4−2(2λ+1) x2)V^{\textrm{QES}}(x) = \tfrac{1}{2}\,(x^6 + 2x^4 - 2(2\lambda +1)\,x^{2})VQES(x)=21​(x6+2x4−2(2λ+1)x2) V QES ( x ) = 1 2 ( x 6 + 2 x 4 - 2 ( 2 λ + 1 ) x 2 ) based on the Wigner quasiprobability. From high-accuracy wavefunctions, we introduce the complex Shannon entropy SWS_WSW​ S W of the Wigner function W ( x , p ), whose imaginary part isolates coherence and negativity, and the entropy S∣W∣S_{|W|}S∣W∣​ S | W | of its modulus, a strictly real indicator of phase-space spreading. Together with the mutual information IWI_WIW​ I W , these measures yield sharp signatures of the transition into the double-well regime ( λ>−12\lambda >-\frac{1}{2}λ>−21​ λ > - 1 2 ) and the emergence of tunneling-induced parity doublets. High-precision spectral data are supported by a smooth Gaussian-process interpolant anchored in asymptotic scaling, ensuring analytic control of level behavior. The resulting phase-space framework provides a compact and sensitive characterization of nonclassical structure beyond probability-density approaches.
In this study, we compare the Wigner function WWW, its modulus, and the Husimi distribution HHH in a one-dimensional quantum system exhibiting a transition from a single-well to a double-well configuration, using the quasi-exactly solvable sextic oscillator as a representative example. High-accuracy variational wavefunctions for the lowest states are used to compute two-dimensional phase-space structures, one-dimensional marginals, and the corresponding Shannon entropies, mutual information, and Cumulative Residual Jeffreys divergences. The analysis shows that the Wigner representation is uniquely responsive to interference effects and displays clear, nonmonotonic entropic behavior as the wells separate, whereas the modulus-Wigner and Husimi distributions account only for geometric splitting or coarse-grained delocalization. These findings establish a quantitative hierarchy in the ability of WWW, ∣W∣|W|∣W∣, and HHH to resolve structural changes in a quantum state and provide a general framework for assessing the descriptive power of different phase-space representations in systems with emerging bimodality or tunneling.
Abstract We present a comparative phase-space analysis of the Wigner function W , the Husimi distribution H , and a normalized auxiliary diagnostic derived from the Wigner function, | W | , for a one-dimensional quantum system undergoing a transition from a single-well to a symmetric double-well potential. Using the quasi-exactly solvable sextic oscillator as a benchmark, we analyze the two-dimensional phase-space structure of these quantities, together with their marginals and associated information-theoretic measures. We show that W uniquely captures phase-space interference, negativity, and the strongest information-theoretic signatures associated with the emergence of the double-well structure, whereas | W | and H primarily highlight geometric splitting and coarse-grained delocalization. In particular, | W | provides a sign-suppressed structural diagnostic that lies between the fully oscillatory Wigner description and the Gaussian-smoothed Husimi picture. These results clarify the distinct information retained by standard and derived phase-space characterizations of the state under continuous deformation of the confining potential.
Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double‐well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation‐induced exact diagonalization procedure. The information entropy‐based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase‐space description provides a semi‐classical explanation for this feature. Total information entropy and phase‐space area show similar behavior with increasing barrier height. image
A theoretical analytical investigation for the exact solvability of non-relativistic quantum spectrum systems, at low energy for modified two-dimensional purely sextic double-well potential (2D-modified sextic DWAO potential) is discussed, by means Bopp's shift method, instead to solving deformed Schrödinger equation with star product, in the framework of both noncommutativite two dimensional real space and phase (NC: 2D-RSP), the exact corrections for lowest excitations are found straightforwardly for interactions in one-electron atoms by means of the standard perturbation theory for three special cases. Furthermore, the obtained corrections of energies are depended on the four infinitesimals parameters (θ, χ) and (θ,σ), which are induced by position-position and momentum-momentum noncommutativity, respectively, in addition to the discreet atomic quantum numbers (j=1±1/2,s=±1/2,1) and the magnetic quantum number m. We have also → generalized our obtained results to include the other atoms with spin S ≠ 1/2.