# Determining Bound States from Møller Operator

Hello I came across an interesting property of the Møller operator, which I summarize below:

The Møller operator $$\Omega^{(+)}$$ maps in-states that belong to the continuum spectrum of the free Hamiltonian $$H_0$$ into states that belong to the continuum spectrum of the exact Hamiltonian $$H$$. For potential scattering, $$H_0$$ has no bound states but $$H$$ may have them. If $$\phi$$ is a bound state of $$H$$ then $$(\phi ,\Omega^{(+)} \psi^{(+)}) = 0$$ for any in-state $$\psi^{(+)}$$.

Using the information given in the answer to my previous post, I tried to look for ways to prove it; however, I could not as I was not sure how the Møller operator acts on eigenstates of the full Hamiltonian (free particle + potential). Can anyone please explain why this property is true?

• Is there a reference for the quote? Where does it come from? Commented Apr 13 at 20:00

If $$H=H_0+V$$, where $$V$$ is a so-called Kato potential, then the point spectrum of $$H$$ corresponds to the bound states, while the scattering states correspond to the continuous spectrum (Ruelle's theorem).

From the spectral theorem, we know that the Hilbert space decomposes as $$\mathscr H=\mathscr H_p \oplus \mathscr H_c$$, which means that bound and scattering states are orthogonal. Since the image of $$\Omega^{(+)}$$ is in $$\mathscr H_c$$, the claim immediately follows.