In mathematics and physics, discrete spectrum of an operator on Hilbert space is the part of the spectrum which corresponds to discrete spectral measures. When defined in this way, it is in fact the closure of the eigenvalues. In the finite dimensional case, the discrete spectrum coincides with the spectrum. In the infinite dimensional case, the discrete spectrum can be empty. The notion, held by some physicists, of "continuous set of eigenvalues" is incorrect. Euclid, detail from The School of Athens by Raphael. ... A Superconductor demonstrating the Meissner Effect. ... In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ... In mathematics, projection-valued measures are used to express results in spectral theory. ...

In quantum mechanics, observables are self adjoint operators, and their spectra are the possible outcomes of the measurement. For example, the position and the momentum operators have purely 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. In physics, momentum is the product of the mass and velocity of an object. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ... In physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. ... It has been suggested that this article or section be merged with Newtonian mechanics. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ...

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. The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... A hydrogen atom is an atom of the element hydrogen. ... // An ion is an atom or group of atoms with a net electric charge. ...