Quadratic forms corresponding to the generalized schrödinger semigroups

Suppose that X_t is the standard Brownian motion in R^d, d ≥ 3, that ρ ∈ H¹(R^d) 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(X_t) - ρ(X₀) - ∫^t₀ ∇ρ(X_s) · dX_s, and let A^μ_t be the continuous additive functional with μ as its Revuz measure. Define A_t as the sum of the two additive functionals above. Then the semigroup defined as T_tf(x) = E^x(e^A_tf(X_t)) is called a generalized Schrödinger semigroup. In this paper we identify the quadratic form corresponding to (T_t) as (E ̃, H¹(R^d)) with E ̃(u, v) = 1 2 ∫ ∇u(x) · ∇v(x) dx + ∫R^d ∇(uv)(x) · ∇ρ(x) dx - ∫_R^du(x) v(x) μ(dx).