correspond to physically impossible outcomes. An example of a quantum mechanical observable which has purely continuous spectrum is the position operator of a free particle moving on a line. Its spectrum is the entire real line. Also, since the momentum operator is unitarily equivalent to the position operator, via the Fourier transform, they have the same spectrum. Intuition may induce one to say that the discreteness of the spectrum is intimately related to the corresponding states being "localized". However, a careful mathematical analysis shows that this is not true. The following is an element of and increasing as . However, the phenomena of Anderson localization and dynamical localization describe, when the eigenfunctions are localized in a physical sense. Anderson Localization means that eigenfunctions decay exponentially as . Dynamical localization is more subtle to define. Sometimes, when performing physical quantum mechanical calculations, one encounters "eigenvectors" that do not lie in L2(R), i.e. wave functions that are not localized. These are the free states of the system. As stated above, in the mathematical formulation, the free states correspond to the absolutely continuous spectrum. Alternatively, if it is insisted that the notion of eigenvectors and eigenvalues survive the passage to the rigorous, one can consider operators on rigged Hilbert spaces. It was believed for some time that singular spectrum is something artificial. However, examples as the almost Mathieu operator and random Schrödinger operators have shown, that all types of spectra arise naturally in physics. Decomposition into essential spectrum and discrete spectrum Let be a closed operator defined on the domain which is dense in X. Then there is a decomposition of the spectrum of A into a disjoint union, where is the fifth type of the essential spectrum of A (if A is a self-adjoint operator, then for all ); is the discrete spectrum of A, which consists of normal eigenvalues, or, equivalently, of isolated points of such that the corresponding Riesz projector has a finite rank. See also Point spectrum, the set of eigenvalues. Essential spectrum, spectrum of an operator modulo compact perturbations. Discrete spectrum (mathematics), the set of normal eigenvalues. Spectral theory of normal C*-algebras Spectrum (functional analysis) References N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience, 1958. M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. Spectral theory