Exact Quasiprobability Hierarchy of the Double-Morse Oscillator: From Potential Geometry to Operator Ordering

Phase-space and quasiprobability methods play key roles across quantum technologies, characterizing localization, non-Gaussianity, nonclassical resources, and coarse-graining. We present a representation-consistent analysis of the lowest quasi-exact ground state of the symmetric double-Morse oscillator. The parameter $A$ changes the potential geometry and physical state, whereas the Cahill–Glauber ordering parameter $s$ changes only the representation and phase-space resolution of a fixed density operator. Although the potential is double-welled for $0<A<1$, the exact ground-state amplitude is single-peaked at the origin and lies above the internal barrier; as $A$ approaches unity, the merged well remains locally quartic rather than harmonic. We obtain closed expressions for the Wigner function and Weyl characteristic function. The Wigner function shows the $A$-dependent exchange between position and momentum localization and retains negative regions, certifying nonclassicality and, for this pure state, non-Gaussianity. The Weyl function gives the Fourier-dual description, generates symmetrically ordered moments and cumulants, and yields the full $s$-ordered hierarchy. For $s<0$, 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 Glauber–Sudarshan $P$ representation remains distributional and does not define a regular positive coherent-state mixture. Thus, the reduced visual extent of Wigner-negative regions as $A$ increases is not classicalization: $A$ controls the physical geometry, while $s$ controls how the same non-Gaussian, nonclassical structure appears across complementary representations. This separation provides an exact benchmark for nonlinear phase-space methods. theory.