Determining Bound States from Møller Operator

Question

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?

Answer 1

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.

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References and further reading:

1. Ruelle, David. "A remark on bound states in potential-scattering theory." Il Nuovo Cimento A (1965-1970) 61.4 (1969): 655-662.

2. Sternberg, Shlomo. A mathematical companion to quantum mechanics. Courier Dover Publications, 2019. Chapter 18.

3. Amrein, Werner O. Hilbert space methods in quantum mechanics. EPFL press, 2009. Chapter 5.