Wigner-entropy and negativity signatures of tunneling in a sextic double well

We develop a phase-space description of tunneling in the quasi-exact sextic double well VQES(x) = 1/2\,(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 SW S W of the Wigner function W ( x , p ), whose imaginary part isolates coherence and negativity, and the entropy S|W| S | W | of its modulus, a strictly real indicator of phase-space spreading. Together with the mutual information IW I W , these measures yield sharp signatures of the transition into the double-well regime ( λ >-1/2 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.