  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
Schr\"odinger-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.

Statistical Mechanics;Exactly Solvable and Integrable Systems;Quantum Physics