For the singular time-dependent perturbations arising in higher-dimensional nonlinear quantum field theory, let H be the free self-adjoint Hamiltonian and let V be the continuous bilinear form on D∞(H), the vectors infinitely differentiable with respect to H. For each nonnegative g∈Cc∞(R), define a distribution propagator relative to H to be a mapping S(g) into the unitary operators on the Hilbert space that belongs to the smallest class containing the propagators generated by bounded self-adjoint perturbations and closed under sequential convergence. Say that the adiabatic perturbation of H by V exists if there is such a distribution propagator for which, for every g, the products S(g/n)n converge as n→∞ to the unitary evolution generated by a self-adjoint operator H(g) whose associated bilinear form extends ∫V(t)g(t)dt.
Determine whether, for the nonlinear quantum-field-theoretic bilinear forms with a compactly supported spatial cutoff f (and subsequently in the limit f→1), the adiabatic perturbation exists uniquely at least as an automorphism of the corresponding Weyl C∗-algebra, with the expected locality and covariance properties, and whether the hyperbolic limit produces an automorphism Σ(g) whose limit as g→1 is unitarily implementable by an essentially unique unitary S-operator.