Faddeev equations

Definition

Consider three particles with the Hilbert space $\mathcal{H} = L^2(\R^6)$ of the relative motion, after removal of the center of mass. Let $H_0$ be the self-adjoint kinetic energy operator. Let $V_1, V_2, V_3$ be the pair potentials, where $V_i$ is the interaction of the pair that does not contain particle $i$. Let $H = H_0 + V_1 + V_2 + V_3$ and $V = V_1 + V_2 + V_3$. For $z \in \C \setminus \R$, let $G_0(z) = (z - H_0)^{-1}$ be the free resolvent. Let $T_i(z)$ be the two-body transition operator of the pair $i$ in the three-body space, defined by $$T_i(z) = V_i + V_i\, G_0(z)\, T_i(z).$$ The Faddeev equations are the system for three operators $T^{(1)}, T^{(2)}, T^{(3)}$ $$T^{(i)}(z) = T_i(z) + T_i(z)\, G_0(z) \sum_{j \ne i} T^{(j)}(z), \qquad i = 1, 2, 3.$$ The sum $T(z) = T^{(1)}(z) + T^{(2)}(z) + T^{(3)}(z)$ then solves the three-body equation $T(z) = V + V\, G_0(z)\, T(z)$. For a bound state $\psi$ of energy $E$ below the spectrum of $H_0$, the wave function form is $\psi = \psi_1 + \psi_2 + \psi_3$ with $$\psi_i = G_0(E)\, T_i(E) \sum_{j \ne i} \psi_j, \qquad i = 1, 2, 3.$$