# What is the Quantum Mechanical analogue of the Bethe-Salpeter equation?

For studying the bound states of quantum fields theories (e.g. studying excitons or mesons), the Bethe-Salpeter equation is often used as the starting point.

Quoting Wikipedia the equation is: $$\Psi=S_1 S_2 K_{12} \Psi$$

Where $$\Psi$$ is the wavefunction being solved for, $$S$$ is the free propagator and $$K$$ is the interaction kernel.

As far as I can understand the equation comes about by assuming the existence of a pole in the Green function (a bound state) and subsequently expanding around this pole to study the bound state.

My question is: is there a quantum mechanical analogue to this equation?, because it really seems like there should be. If such an analogue exists, is it ever used and why or why not?