An arXiv preprint reports that a double-well oscillator can have a lowest quasi-exact ground-state amplitude that remains a single peak at the center, even though the potential has two wells. In the stated regime, A is greater than 0 and less than 1.
Two parameters play different roles. A indexes different potentials and corresponding physical states, while s indexes operator ordering and phase-space resolution for one fixed density operator. In practical terms, A selects the modeled geometry and state; s selects how that fixed state is represented.
One peak in a two-well landscape
The result is not a two-peak ground state hidden inside the double well. For the range where A lies between 0 and 1, the lowest quasi-exact ground-state amplitude is even and single-peaked at the central coordinate.
The exact state lies above the internal barrier rather than forming a conventional below-barrier tunnelling doublet. When the two minima coalesce, the central well remains locally quartic instead of approaching a harmonic-oscillator limit.
The same state, different maps
To examine the same state in phase space, a map that tracks coordinate and momentum together, the work derives closed analytical Wigner and Weyl characteristic functions. The Weyl function is used to generate symmetrically ordered moments, cumulants and the full s-ordered hierarchy. The resulting descriptions are different representations of one fixed density operator, not separate physical states.
At larger A, the displayed comparisons show a narrower central Wigner-positive region along the coordinate axis q and a broader region along the momentum axis p. The visible negative lobes become less extensive but remain. The comparison is qualitative and based on selected displayed cases.
The paper interprets the negative Wigner regions as witnesses of both nonclassicality and non-Gaussianity. It does not interpret the reduced visible extent of the negative lobes at larger A as a transition to classical behavior.
Smoothing changes the view
The ordering parameter s marks different levels of representational resolution. For negative s, isotropic Gaussian smoothing suppresses fine Wigner-negative structure while preserving the dominant localization envelope. At the Husimi endpoint, labeled s = -1, the Husimi distribution is nonnegative.
The Husimi function is defined from the diagonal coherent-state matrix element of the density operator, meaning the matrix element with the same coherent state on both sides, divided by pi. It is a smoothed representation of the same fixed state, not a new state.
At the opposite endpoint, s = 1, the inverse transform defining the Glauber-Sudarshan P representation is distributional rather than an ordinary integrable function. It must be interpreted as a distribution rather than read as an ordinary curve.
A focused calculation
The analysis is confined to the lowest quasi-exact sector, with n equal to zero and the dimensionless parameter mu equal to one. The comparisons therefore describe this sector's state across the model's representations.
The displayed comparison at larger A is qualitative and based on selected cases. The selected s values illustrate the smoothing hierarchy but do not determine a numerical nonclassical-depth threshold.
Front matter identifies the work as arXiv:2608.23316v3, dated 29 August 2026. The plot data were generated by direct implementation of the equations and parameters in the text and are available from the corresponding author upon reasonable request.
Paper data and sources
Original title: Exact Quasiprobability Hierarchy of the Double-Morse Oscillator: From Potential Geometry to Operator Ordering
Authors: F. Chogle, B. Teklu, M. F. Pereira
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-24
DOI: Not available
Original paper · Full text