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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?

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