In quantum mechanics the scattering matrix is often used, to connect the states on the left and right side over some region in the middle. In the trivial case, that the Hamiltonian in the middle is the same as left and right, there is no reflection and the transmission probability for each channel is $1$. Now this probability is independent of energy of course.

In my case, I am interested in an energy independent scattering matrix. I wonder, is there some kind of well defined set of Hamiltonians, that leads to such an energy-independent scattering matrix?

The reason for why I am asking this, is the Landauer formula that relates the conductance of a sample to the transmission probability.

$$G = \frac{e^2}{\pi\hbar} \sum_n T_n $$ Here $n$ is an index describing different channels.

However, this is only valid, if the transmission probability is independent of energy, bringing me back to my question. When are Transmission probabilities independent of energy?