  We consider a one-channel coherent conductor with a good transmission
embedded into an ohmic environment whose impedance is equal to the quantum of
resistance R_q=h/e^2 below the RC frequency. This choice is motivated by the
mapping of this problem to a Tomonaga-Luttinger liquid with one impurity whose
interaction parameter corresponds to the specific value K=1/2, allowing for a
refermionization procedure. The "new" fermions have an energy-dependent
transmission amplitude which incorporates the strong correlation effects and
yields the exact dc current and zero-frequency noise through expressions
similar to those of the scattering approach. We recall and discuss these
results for our present purpose. Then we compute, for the first time, the
finite-frequency differential conductance and the finite-frequency
non-symmetrized noise. Contrary to intuitive expectation, both cannot be
expressed within the scattering approach for the new fermions, even though they
are still determined by the transmission amplitude. Even more, the
finite-frequency conductance obeys an exact relation in terms of the dc current
which is similar to that derived perturbatively with respect to weak tunneling
within the Tien-Gordon theory, and extended recently to arbitrary strongly
interacting systems coupled eventually to an environment or/and with a
fractional charge. We also show that the emission excess noise vanishes exactly
above eV, even though the underlying Tomonaga-Luttinger liquid model
corresponds to a many-body correlated system. Our results apply for all ranges
of temperature, voltages and frequencies below the RC frequency, and they allow
to explore fully the quantum regime.
