Exact Cahill–Glauber hierarchy for the double-Morse ground state

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 A controls the separation of the minima and the central barrier, thereby changing the physical ground state. At fixed A, the Cahill–Glauber parameter s labels the quasiprobability WA(s)(q,p): s = 0, −1, and 1 give the Wigner, Husimi Q, and Glauber–Sudarshan P representations, respectively. 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 barrier; as A 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 A-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 s-ordered hierarchy. For s < 0, isotropic Gaussian smoothing suppresses fine sign-changing structure while preserving the large-scale localization envelope. The Husimi end point is non-negative without implying classicality, whereas the P representation remains distributional. Thus, A controls the physical phase-space geometry, while s controls how the same non-Gaussian and nonclassical state is resolved across complementary representations.