  The saddle point approximation of the path integral partition functions is an
important way of deriving the thermodynamical properties of black holes.
However, there are certain black hole models and some mathematically analog
mechanical models for which this method cannot be applied directly. This is due
to the fact that their action evaluated on a classical solution is not finite
and its first variation does not vanish for all consistent boundary conditions.
These problems can be dealt with by adding a counterterm to the classical
action, which is a solution of the corresponding Hamilton-Jacobi equation.
  In this work we study the effects of polymer quantization on a mechanical
model presenting the aforementioned difficulties and contrast it with the above
counterterm method. This type of quantization for mechanical models is
motivated by the loop quantization of gravity which is known to play a role in
the thermodynamics of black hole systems.
  The model we consider is a nonrelativistic particle in an inverse square
potential, and analyze two polarizations of the polymer quantization in which
either the position or the momentum is discrete. In the former case, Thiemann's
regularization is applied to represent the inverse power potential but we still
need to incorporate the Hamilton-Jacobi counterterm which is now modified by
polymer corrections. In the latter, momentum discrete case however, such
regularization could not be implemented. Yet, remarkably, owing to the fact
that the position is bounded, we do not need a Hamilton-Jacobi counterterm in
order to have a well-defined saddle point approximation. Further developments
and extensions are commented upon in the discussion.

General Relativity and Quantum Cosmology;High Energy Physics - Theory;Quantum Physics