Defining Hamiltonian density and deriving energy current

[A. Neumaier]

Great. What do you think about my treatment of the external potential in my post #28 ? The one-body potential V(r) always disappears for the particle current, so does it also disappear for the energy current?

No; one gets contributions from the spatial derivatives of the interaction potential. I haven't time to look at the details of your derivation but an explicit formula is given at the end of Section 7H ''Microscopic balance equations (quantum fluids)'' of Reichl's Modern Course in Statistical Physics.