Hamiltonian Feynman measures, Kolmogorov integral and infinite dimensional pseudo-differential operators

Properties of infinite dimensional pseudo-differential operators (PDO) are discussed; in particular, the connection between two definitions of the PDO is considered: one given in terms of the Hamiltonian Feynman measure, and another introduced in this work in terms of the Kolmogorov integral.

A path integral approach to disordered systems

The exact expression for the average propagator of a completely disordered system is evaluated by using a method of expansion in terms of cumulants defined over the Feynman measure. The expansions are evaluated for a one-dimensional model of delta function potentials and for a three-dimensional screened Coulomb potential to give the asymptotic form of the density of states for | E | →∞. Using a functional Taylor expansion it is shown that the average propagator may be expanded to give approximate non-linear integral equations for either the average propagator or average Green function. Using the cumulant formulation it is shown that an effective quantum potential may be defined in terms of which the propagator may be calculated.