Continuous vs discrete set of eigenvectors for a single particle Hamiltonian

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?

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.
• @biryani Yes there is a result from functional analysis about the decay rate of the solution in the region where $E< V$. – ZeroTheHero Mar 23 '17 at 14:15