eigenfunction

English

Etymology

From eigen- +‎ function.

Noun

eigenfunction (plural eigenfunctions)

1. (mathematics) A function $\phi$ such that, for a given linear operator $D$, $D\phi =\lambda \phi$ for some scalar $\lambda$ (called an eigenvalue).
   • 2000, John von Neumann, E. Wigner, “On the Behaviour of Eigenvalues in Adiabatic Processes”, in Hinne Hettema, transl., edited by Hinne Hettema, Quantum Chemistry: Classic Scientific Papers, World Scientific, published 1929, page 25:

     For many quantum-mechanical problems it is important to investigate the change of eigenvalues and eigenfunctions with the continuous change of one or more parameters. The case in which one knows the eigenvalues and eigenfunctions for two special values of the parameters, and is interested in the region in between is particularly interesting.

   • 2005, Leonid D. Akulenko, Sergei V. Nesterov, High-Precision Methods in Eigenvalue Problems and Their Applications‎^[1], CRC Press (Chapman & Hall), page 1:

     Problems that require an investigation of eigenvalues and eigenfunctions arise in connection with numerous topics in mechanics, the theory of vibrations and stability, hydrodynamics, elasticity, acoustics, electrodynamics, quantum mechanics, etc.

Synonyms

• proper function, eigenstate

Related terms

• eigenbasis
• eigenbrain
• eigendecomposition, eigen decomposition
• eigenface
• eigenhead
• eigenmode
• eigenpair
• eigenspace
• eigenspinor
• eigenstate
• eigensystem
• eigentheory
• eigenvalue
• eigenvector

Translations

function

• Arabic: ^{(please verify)} دَالَة خَاصَّة f (dāla ḵāṣṣa)
• Chinese:

  Mandarin: 本徵函數／本征函数 (běnzhēng hánshù)

• Finnish: ominaisfunktio
• French: fonction propre (fr) f
• German: Eigenfunktion (de) f
• Greek: ιδιοσυνάρτηση f (idiosynártisi)
• Icelandic: eiginfall n
• Italian: autofunzione (it) f
• Japanese: 固有関数 (ja) (koyū kansū)
• Portuguese: autofunção f (Brazil), função própria f (Portugal)
• Russian: со́бственная фу́нкция f (sóbstvennaja fúnkcija)
• Spanish: autofunción f, función propia f
• Swedish: egenfunktion (sv) c

See also

• hamiltonian

Further reading

• Mathworld article on eigenfunctions