Functional Determinants in the Stationary Action Approximation

$$ \frac{\mathrm{Det}O_{\mathcal{A},t_{\mathfrak{0}},\mathfrak{t_{1}}}^{(u_{\mathfrak{1}})} }{\mathrm{Det}O_{\mathcal{A},t_{\mathfrak{0}},\mathfrak{t_{1}}}^{(u_{\mathfrak{1}})}}=\frac{\mathrm{det}(A_{\mathfrak{0}}+A_{\mathfrak{1}}\Phi_{*\, t_{\mathfrak{0}},\mathfrak{t_{1}}}^{(u_{\mathfrak{1}})})}{\mathrm{det}(A_{\mathfrak{0}}+A_{\mathfrak{1}}\Phi_{*\, t_{\mathfrak{0}},\mathfrak{t_{1}}}^{(u_{\mathfrak{1}})})} $$ A beautiful result by Robin Forman yields a powerful expression of functional determinants occurring in elliptic boundary value problems.
Functional determinants occurring in the stationary action approximation in finite dimensional Quantum Mechansics and non-equilibrium Statistical Mechanics admit an explicit expression for any elliptic boundary conditions.
In my Physics Report , I used Forman's result to present a new derivation of Gutzwiller's trace formula. In my view functional determinant theory offers a particularly appealing way to understand topological invariance of Masolv and Conley and Zehnder indices in periodic orbit theory.