A parametrized double-well potential is proposed to address the issue of the impact of shape deformability of some bistable physical systems, on their quantum dynamics and classical statistical mechanics. The parametrized double-well potential possesses two fixed degenerate minima and a constant barrier height, but a tunable shape of its walls influencing the confinement of the two symmetric wells. The transition from quantum tunneling to classical crossover is investigated and it is found that unlike the bistable model based on the Ginzburg-Landau energy functional, so-called $phi^4$ potential model, members of the family of parametrized double-well potentials can display a first-order transition above a universal critical value of the shape deformability parameter. Addressing their statistical mechanics within the framework of the transfer-integral operator formalism, the classical partition function is constructed with emphasis on low-lying eigenstates of the resulting Schrodinger-type equation. The quasi-exact solvability of this last equation to find the exact expression of the partition function, is shown to depend on a specific relationship between the shape deformability parameter and the thermodynamic temperature. The quasi-exact solvability condition is expressed analytically, and with this condition some exact wavefunctions and corresponding eigenenergies are derived at several temperatures. The exact probability densities obtained from the analytical expressions of the groundstate wavefunctions at different temperatures, are found to be in excellent agreement with the probability density from the Langevin dynamics. This agreement makes it possible to study probability density-based thermodynamics via Langevin techniques.
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