Integral representations for the Schroedinger propagator

We consider the Schro ̈dinger equation for a Hamiltonian operator with a potential function modeling one-particle scattering problems. By means of a strongly converging regularization of the Schro ̈dinger 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 related to the group of classical canonical transformations. Moreover, as a particular generating function, we consider the action functional A[γ] evaluated on a suitable finite-dimensional space of curves γ ∈ Ŵ ⊂ H1([0,t],Rn). As a matter of fact we obtain a finite-dimensional path integral representation for the relaxed kernel.