Path integral for half-binding potentials as quantum mechanical analog for black hole partition functions

Understand

The semi-classical approximation to black hole partition functions is not well-defined, because the classical action is unbounded and the first variation of the uncorrected action does not vanish for all variations preserving the boundary conditions.

• Both problems can be solved by adding a Hamilton-Jacobi counterterm.
• I show that the same problem and solution arises in quantum mechanics for half-binding potentials.