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,
Is there any potential that can give rise to an infinite and continuous set of eigenvectors (barring the uniform potential)?
If not, why should the addition of a potential change the eigenspectrum from discrete to continuous?
If the answer to the first question is yes, what conditions should the potential satisfy to have a discrete spectra?