Quadratic Forms Corresponding to the Generalized Schrِdinger Semigroups

Suppose that Xt is the standard Brownian motion in Rd, d ≥ 3, that ρ ∈ H1(Rd) is a bounded continuous function such that |∇ρ|2 belongs to the Kato class and μ is a measure belonging to the Kato class. Let A[ρ]t be defined as A[ρ]t = ρ(Xt) − ρ(X0) − ∫t0 ∇ρ(Xs) · dXs, and let Aμt be the continuous additive functional with μ as its Revuz measure. Define At as the sum of the two additive functionals above. Then the semigroup defined as Ttf(x) = Ex{eAtf(Xt)} is called a generalized Schrödinger semigroup. In this paper we identify the quadratic form corresponding to (Tt) as (Ẽ, H1(Rd)) with Ẽ(u, v) = 12 ∫ ∇u(x) · ∇v(x) dx + ∫Rd ∇(uv)(x) · ∇ρ(x) dx − ∫Rdu(x) v(x) μ(dx).