The (quadratic) Hamiltonian is H ( λ λ ) = 1 2 κ κ ( L ( λ λ ) , L ( λ λ ) ) which is indeed a function on the phase space, which is additionally dependent on a spectral parameter λ λ . This can be written as H ( λ λ ) = Δ Δ ∞ ∞ + ∑ ∑ r = 1 N ( Δ Δ r ( λ λ − − λ λ r ) 2 + H r λ λ − − λ λ r ) , with Δ Δ r = 1 2 κ κ ( X ( r ) , X ( r ) ) , Δ Δ ∞ ∞ = 1 2 κ κ ( Ω Ω , Ω Ω ) and H r = ∑ ∑ s ≠ ≠ r κ κ ( X ( r ) , X ( s ) ) λ λ r − − λ λ s + κ κ ( X ( r ) , Ω Ω ) .