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The Hamiltonian operator of a free particle in three dimensional Euclidean space has an infinite set of eigenvectors lableled by the momentum of the particle, $| p \rangle $. Not only is this set infinite, but it is also continuous. But if we look at exactly solvable Hamiltonians with a potential, like the harmonic oscillator or the $1/r^2$ potential, the set of eigenvectors form a infinite but discrete set (labelled by a set of integral quantum numbers). I have two questions regarding this,

  1. Is there any potential that can give rise to an infinite and continuous set of eigenvectors (barring the uniform potential)?

  2. If not, why should the addition of a potential change the eigenspectrum from discrete to continuous?

  3. If the answer to the first question is yes, what conditions should the potential satisfy to have a discrete spectra?

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Whether a particular Hamiltonian gives rise to discrete or continuous momentum spectra depends on the energy of the particle(assuming the particle is in energy eigen-state). For instance consider an electron in Hydrogen atom potential but with energy $E > 0$. Then the electrons motion is unbounded and it can have continuous $|p>$ eigenstates.

A more interesting question could be that given a bounded system is it possible to have continuous momentum spectra. The answer again is YES. In crystalline solids the electron energy levels are not of specific momentum but rather a "continuous band of energy" at discrete intervals.

To have a perfect discrete spectra your potential should have infinities at some boundary. This ensures that the wave-function has to be zero at some boundary and therefor be necessarily discrete.

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    $\begingroup$ Actually this isn't strictly quite correct (and an occasional source of confusion) as the energy spectrum of the harmonic oscillator is strictly positive. The key point is whether the state is bound or not. See section 6 of this older but well written paper: tandfonline.com/doi/abs/10.1080/00107516408203103# The crystal example is also a source of confusion as the wavefunctions have very large spatial extend (by Bloch's theory they are in fact periodic). $\endgroup$ – ZeroTheHero Mar 23 '17 at 13:46
  • $\begingroup$ Sorry, I'm bit confused by your comment. Can you elaborate on what part you think is not correct? $\endgroup$ – Ari Mar 23 '17 at 13:48
  • $\begingroup$ Doesn't the Harmonic oscillator have positive energy and still a discrete spectrum? $\endgroup$ – biryani Mar 23 '17 at 14:01
  • $\begingroup$ Yes. But my positive energy condition was only for Hydrogen atom potential. I meant that for that particular potential if E>0 then you get continuous spectra. Sorry if that's not clear. $\endgroup$ – Ari Mar 23 '17 at 14:04
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    $\begingroup$ @biryani Yes there is a result from functional analysis about the decay rate of the solution in the region where $E< V$. $\endgroup$ – ZeroTheHero Mar 23 '17 at 14:15

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