Interactions along Brownian paths in Rd, d≤5

Via the Posilicano method, Schroedinger operators with singular potentials supported by Brownian paths in the configuration space Rd, 1≤d≤5, are constructed. The essential, absolutely continuous and singular continuous spectra are determined almost surely (with respect to Wiener measure). It is shown that the set of positive eigenvalues is discrete and that the wave operators exist and are asymptotically complete almost surely; if d ≥ 3 then the set of positive eigenvalues is even empty almost surely. A trace formula for the number (counting multiplicities) of negative eigenvalues is derived