Discrete spectrum
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In physics, discrete spectrum is a finite set or a countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a discrete spectrum if its eigenvalues cannot be changed continuously. If the spectrum of an operator is not discrete, we say that it is a continuous spectrum.
The position and the momentum operators (in an infinite space) have continuous spectra. But the momentum in a compact space, the angular momentum, and the Hamiltonian of various physical systems (the corresponding eigenstates are called bound state) have a discrete (quantized) spectrum. This is a major difference with the corresponding operators in classical mechanics. Quantum mechanics was therefore named in this way.
The quantum harmonic oscillator or the Hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the Hydrogen atom, it has both continuous as well as discrete part of the spectrum; the continuous part represents the ionized atom.
For a mathematically rigorous point of view, see decomposition of spectrum (functional analysis).
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