Sturm–Liouville problem

E697758

boundary value problem concept in differential equations eigenvalue problem

The Sturm–Liouville problem is a class of second-order linear differential equations with boundary conditions that yield real eigenvalues and orthogonal eigenfunctions forming a basis for function expansions in mathematical physics and engineering.

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All labels observed (7)

Label	Occurrences
Sturm–Liouville theory	3
Eigenfunction Expansions Associated with Second-order Differential Equations	1
Sturm–Liouville operator	1
Sturm–Liouville operators	1
Sturm–Liouville problem canonical	1
Sturm–Liouville problems	1
Sturm–Liouville theory of eigenfunction expansions	1

How this entity was disambiguated

This entity first appeared as the object of triple T7871809 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Sturm–Liouville problem
Context triple: [Jacobi polynomials, areSolutionsOf, Sturm–Liouville problem]

A. Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
B. Gelfand–Levitan theory
Gelfand–Levitan theory is a foundational framework in inverse spectral theory that reconstructs differential operators or potentials from their spectral data using integral equations.
C. Cauchy–Euler equation
The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
D. Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
E. Poisson equation
The Poisson equation is a fundamental partial differential equation in mathematical physics that relates the Laplacian of a potential field to a given source distribution, widely used in electrostatics, gravitation, and heat conduction.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Sturm–Liouville problem
Target entity description: The Sturm–Liouville problem is a class of second-order linear differential equations with boundary conditions that yield real eigenvalues and orthogonal eigenfunctions forming a basis for function expansions in mathematical physics and engineering.

A. Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
B. Gelfand–Levitan theory
Gelfand–Levitan theory is a foundational framework in inverse spectral theory that reconstructs differential operators or potentials from their spectral data using integral equations.
C. Cauchy–Euler equation
The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
D. Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
E. Poisson equation
The Poisson equation is a fundamental partial differential equation in mathematical physics that relates the Laplacian of a potential field to a given source distribution, widely used in electrostatics, gravitation, and heat conduction.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	boundary value problem ⓘ concept in differential equations ⓘ eigenvalue problem ⓘ
assumesConditionOn	p(x) > 0 on (a,b) ⓘ w(x) > 0 on (a,b) ⓘ
definedOnInterval	[a,b] ⓘ
hasBoundaryConditions	linear homogeneous boundary conditions at x=a and x=b ⓘ
hasBoundaryConditionType	Dirichlet boundary conditions ⓘ Neumann boundary conditions ⓘ linked to: Neumann boundary conditions in potential theory Robin boundary conditions ⓘ periodic boundary conditions ⓘ
hasCoefficientFunction	p(x) ⓘ q(x) ⓘ
hasEigenvalueEquation	L[y] = λ w(x) y(x) ⓘ
hasGeneralForm	-(d/dx)[p(x) y'(x)] + q(x) y(x) = λ w(x) y(x) ⓘ
hasHistoricalDevelopmentPeriod	19th century ⓘ
hasKeyResult	Sturm–Liouville theory of eigenfunction expansions ⓘ linked to: Sturm–Liouville problem
hasKeyTheorem	Sturm comparison theorem ⓘ Sturm oscillation theorem ⓘ Sturm separation theorem ⓘ
hasOperator	L[y] = -(d/dx)[p(x) y'(x)] + q(x) y(x) ⓘ
hasOrder	second-order ⓘ
hasOrthogonalityRelation	∫_a^b w(x) y_m(x) y_n(x) dx = 0 for m ≠ n ⓘ
hasProperty	eigenfunctions form a complete set under suitable conditions ⓘ eigenfunctions form an orthogonal set with respect to w(x) ⓘ real eigenvalues ⓘ self-adjoint differential operator ⓘ
hasSpecialCase	Bessel differential equation ⓘ linked to: Bessel functions Hermite differential equation ⓘ Laguerre differential equation ⓘ Legendre differential equation ⓘ linked to: Legendre’s differential equation spherical harmonics eigenvalue problem ⓘ
hasSpectrum	discrete under regular boundary conditions ⓘ
hasType	linear differential equation ⓘ
hasWeightFunction	w(x) ⓘ
namedAfter	Jacques Charles François Sturm ⓘ linked to: Charles-François Sturm Joseph Liouville ⓘ
relatedTo	Hilbert space theory ⓘ orthogonal polynomials ⓘ spectral theory of linear operators ⓘ
usedFor	Fourier-type series expansions ⓘ separation of variables in partial differential equations ⓘ
usedIn	engineering ⓘ heat conduction problems ⓘ mathematical physics ⓘ quantum mechanics ⓘ vibrations and acoustics ⓘ

How these facts were elicited