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We compute both analytically and numerically the geometry of the parameterspace of the anharmonic oscillator employing the quantum metric tensor and itsscalar curvature. A novel semiclassical treatment based on a Fourierdecomposition allows to construct classical analogues of the quantum metrictensor and of the expectation values of the transition matrix elements. Adetailed comparison is presented between exact quantum numerical results, aperturbative quantum description and the semiclassical analysis. They are shownto coincide for both positive and negative quadratic potentials, where thepotential displays a double well. Although the perturbative method is unable todescribe the region where the quartic potential vanishes, it is remarkable thatboth the perturbative and semiclassical formalisms recognize the negativeoscillator parameter at which the ground state starts to be delocalized in twowells.

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