I Deduce if the spectrum is discrete/continuous from the potential

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Summary
I have one-dimensional problem with a one-dimensional potential
I want to know the energy domains that will result in discrete energy levels and the energy domains that will result in continuous energy levels
I have one-dimensional problem with a one-dimensional potential
I want to know the energy domains that will result in discrete energy levels and the energy domains that will result in continuous energy levels

In my lecture, my professor gave the example of v(r) = 1/r (r>0) (hydrogen atom basically). he told us that we can know immediately that for E>0 we will get continuous spectrum and for E<0 discrete spectra.
As far as I understood him, its because when r goes to infinity V goes to 0.

I would like to know the full explanation and why it works (some sort of proof would be nice ).
I tried to look for a decent explanation in Sakurai ( the coursebook) but unfortunately, I couldn't find one
 

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Summary: I have one-dimensional problem with a one-dimensional potential
I want to know the energy domains that will result in discrete energy levels and the energy domains that will result in continuous energy levels

I have one-dimensional problem with a one-dimensional potential
I want to know the energy domains that will result in discrete energy levels and the energy domains that will result in continuous energy levels

In my lecture, my professor gave the example of v(r) = 1/r (r>0) (hydrogen atom basically). he told us that we can know immediately that for E>0 we will get continuous spectrum and for E<0 discrete spectra.
As far as I understood him, its because when r goes to infinity V goes to 0.

I would like to know the full explanation and why it works (some sort of proof would be nice ).
I tried to look for a decent explanation in Sakurai ( the coursebook) but unfortunately, I couldn't find one
If you assume for simplicity that the potential goes to ##0## at infinity, then ##E < 0## results in bound states that have a discrete spectrum. And ## E > 0## results in scattering states that typically have a continuous spectrum.

You could search online for a proof of this.
 
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Interestingly enough, there are actually systems where positive energy bound states are emebedded in the continuous spectrum. See ballentine.
 

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