  The Hamiltonian of electrons interacting with interface phonons in
three-barrier resonant tunneling structure is established using the first
principles within the models of effective mass and polarization continuum.
Using the Green's functions method, the temperature shifts and decay rates of
operating electron states are calculated depending on geometric design of
three-barrier nano-structure GaAs/Al$_{x}$Ga$_{1-x}$As which is an active
region of quantum cascade detector. It is established that independently of the
temperature, the energy of quantum transition during the process of
electromagnetic field absorption is a nonlinear weakly varying function of the
position of the inner barrier with respect to the outer barriers of the
structure.
