Functions of unitary operators: Derivatives and trace formulas

Let A be a bounded self-adjoint operator on a separable Hilbert space H and U0 an arbitrary unitary operator. We derive formulas for higher order derivatives of the function s→φ(eisAU0) for sufficiently smooth functions φ. We compare our formulas with earlier results in this area and apply them to establish higher order analogs of trace formulas of Krein-Koplienko-Neidhardt.