Interactions along Brownian paths: Completeness and eigenvalues

Let 0 < c, T < ∞ and for every ω ∈ C(ℝ+, ℝ3) let µTω be the occupation time measure of ω until time T. Let W be the Wiener measure. W-a.s. the wave operators W±:(−Δ −cµTω, −Δ) exist and are asymptotically complete. The expectation value (with respect to W) of the number, counting multiplicities, of negative eigenvalues of −Δ −cµTω is finite. On the other hand, for every N ∈ ℕ and E0 ∈ ℝ the probability that −Δ −cµTω has more than N eigenvalues below E0 is positive.