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Journal of Physics A Mathematical and Theoretical

Wigner–Husimi phase-space structure of quasi-exactly solvable sextic potential

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

Published Jun 22, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/ae8089

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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.
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.
Phase-space and quasiprobability methods now play operational roles in quantum technologies, characterizing localization, non-Gaussianity, nonclassical resources, and coarse-graining. We develop an exact, representation-consistent analysis of the lowest quasi-exact ground state of the symmetric double-Morse oscillator. In the double-Morse potential, the dimensionless parameter AAA controls the separation of the minima and the central barrier, thereby changing the physical ground state. At fixed AAA, the Cahill--Glauber parameter sss labels the quasiprobability WA(s)(q,p)W_A^{(s)}(q,p)WA(s)​(q,p): s=0s=0s=0, −1-1−1, and 111 give the Wigner, Husimi QQQ, and Glauber--Sudarshan PPP representations, respectively. Although the potential is double-welled for 0<A<10<A<10<A<1, the exact ground-state amplitude is single-peaked at the origin and lies above the barrier; as AAA approaches unity, the merged well remains locally quartic rather than harmonic. Closed analytical expressions are obtained for the Wigner function and Weyl characteristic function. The Wigner function displays the AAA-dependent exchange between position and momentum localization and retains negative regions, certifying nonclassicality and, for this pure state, non-Gaussianity. The Weyl function is its Fourier dual, generates symmetrically ordered moments and cumulants, and yields the full sss-ordered hierarchy. For s<0s<0s<0, isotropic Gaussian smoothing suppresses fine sign-changing structure while preserving the large-scale localization envelope. The Husimi endpoint is nonnegative without implying classicality, whereas the PPP representation remains distributional. Thus, AAA controls the physical phase-space geometry, while sss controls how the same non-Gaussian and nonclassical state is resolved across complementary representations
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.
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.
We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter AAA, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.
The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.