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Discrete means as opposed to continuous. For instance, people may ask questions about discrete electric charges, discrete spacetime, discrete energies, etc. If discretization is vital/essential to the question, then tag it with this tag.
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Continuous vs discrete set of eigenvectors for a single particle Hamiltonian
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? …