March-Russell, John Theory Division, European Center for Nuclear Research, Geneva, Switzerland.
Last reviewed:January 2020
- Matrix equations
- Linear operators
- Differential eigenvalue equations
- Related Primary Literature
- Additional Reading
One of the solutions of an eigenvalue equation. A parameter-dependent equation that possesses nonvanishing solutions only for particular values (eigenvalues) of the parameter is an eigenvalue equation, the associated solutions being the eigenfunctions (sometimes eigenvectors). In older usage the terms characteristic equation and characteristic values (functions) are common. Eigenvalue equations appear in many contexts, including the solution of systems of linear algebraic equations (matrix equations), differential or partial differential equations, and integral equations. The importance of eigenfunctions and eigenvalues in applied mathematics results from the widespread applicability of linear equations as exact or approximate descriptions of physical systems. However, the most fundamental application of these concepts is in quantum mechanics where they enter into the definition and physical interpretation of the theory. Only linear eigenvalue equations will be discussed. See also: Eigenvalue (quantum mechanics); Energy level (quantum mechanics); Nonrelativistic quantum theory; Quantum mechanics
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