Parameter space geometry of the quartic oscillator and the double-well potential: classical and quantum description

We compute the quantum metric tensor and its scalar curvature for the anharmonic oscillator for positive and negative quadratic potentials, where the potential displays a double well, employing exact numerical and perturbative procedures. We also introduce a formulation of the classical analog of the quantum metric tensor by using a novel approach based on Fourier series, which is shown to reproduce the relevant quantum features involved in the parameter space. It is remarkable that both the exact quantum treatment and classical formalism recognize the negative oscillator parameter at which the ground state starts to be delocalized in two wells.