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Sturm–Liouville theory — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-06T18:06:54.440Z
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Sturm–Liouville theory — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_b0fee0ffee99 (Metraipolis archive dc_b0fee0ffee99, sha256 15874097d58e25470e382aa2dcf5e48526440fd5b3a7a4883b3c0e9f4eb4c7fb)

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@misc{dcb0fee0ffee99,
  title = {Sturm–Liouville theory — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_b0fee0ffee99; sha256 15874097d58e25470e382aa2dcf5e48526440fd5b3a7a4883b3c0e9f4eb4c7fb},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form d d x [ p ( x ) d y d x ] + q ( x ) y = − λ w ( x ) y {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y} for given functions p ( x ) {\displaystyle p(x)} , q ( x ) {\displaystyle q(x)} and w ( x ) {\displaystyle w(x)} , together with some boundary conditions at extreme values of x {\displaystyle x} . The goals of a given Sturm–Liouville problem are: To find the λ {\displaystyle \lambda } for which there exists a non-trivial solution to the problem. Such values λ {\displaystyle \lambda } are called the eigenvalues of the problem. For each eigenvalue λ {\displaystyle \lambda } , to find the corresponding solution y = y ( x ) {\displaystyle y=y(x)} of the problem. Such functions y {\displaystyle y} are called the eigenfunctions associated to each λ {\displaystyle \lambda } . Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions. This theory is important in applied mathematics, where Sturm–Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations. For example, in quantum mechanics, the one-dimensional time-independent Schrödinger equation is a Sturm–Liouville problem. Sturm–Liouville theory is named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), who developed the theory.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «differential equation»).
Fonte: Wikipedia in inglese, voce «Sturm–Liouville theory» — https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory
Wikidata Q1154082 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q1154082
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1351213655, consultata il 2026-10-06T16:32:14.372Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: ba919ae357b1be1a8bab6a63e872491ea806435e10b924bd3103e29ec36e3188