The "dressed" states have an energy E n ′ = d n + ∑ ∑ k V n k ′ V k n ′ d n − − d k following the recipe for first-order (non-degenerate) perturbation theory. This is applicable since the SW transformation is based on the approximation | V i j | ≪ ≪ | d i − − d j | Note that the unitary rotation U does not affect the eigenvalues, meaning that E n ′ is also a meaningful approximation for the original Hamiltonian H .