Integral representations of the schrödinger propagator

We consider the Schrödinger equation for the Hamiltonian operator H = −ħ2/2m + Δ + V (x), where V is a potential function modeling one-particle scattering problems. By means of a strongly converging regularization of the Schrödinger propagator U(t), we introduce a new class of integral representations for the relaxed kernel in terms of oscillatory integrals. They are constructed with complex amplitudes and real phase functions that belong to the class of global weakly quadratic generating functions of the Lagrangian submanifolds Λt ⊂ T★ℝn × T★ℝn related to the group of classical canonical transformations ølH. Moreover, as a particular generating function, we consider the action functional A[γ] = ∫0t ½ m ׀ẏ(s)׀2 − V(γ y(s))ds evaluated on a suitable finite-dimensional space of curves γ ∈ Γ ⊂ H1 ([0, t],ℝn). As a matter of fact we obtain a finite-dimensional path integral representation for the relaxed kernel.