Resonances, Lagrangians and Quantum Propagator/Partition Function

In classical mechanics, it is possible to have different Lagrangians yield identical equations of motion. For example, a velocity dependent potential x dx/dt does not affect the equations of motion, but leads to a different definition for momentum i.e. p= mdxd/dt - x versus p = mdx/dt. (In general, dG(x)/dt behaves in a similar manner.) In quantum mechanics, p→ -i d/dx and so mdxd/dt - x is the momentum used. Given a bound state wavefunction W(x) written as a Fourier series, it is the new momentum which is represented by p. This p which is not mdx/dt is involved in forming quantum resonances i.e. solutions of the Schrodinger equation. These resonances, however, do not necessarily affect the quantum partition function because the extra potential may create a phase, similar to the result of writing the Schrodinger equation viewed from a moving frame. Thus, one may have two different Hamiltonians physically equivalent to: m/2 dx/dt dx/dt + V(x) namely: p/2m + V(x) where p=mdx/dt and (p+x)(p+x)/2 + V(x) where p=mdx/dt - x. In this note, we examine the partition function from the Feynman path integral approach. Using this propagator approach one may find different partition functions for each case, even though E= m/2 dx/dt dx/dt + V(x). Setting X=Y, however, yields a P(x) which is the same. We consider the case of an extra potential of d/dt G(x)