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https://www.physicsforums.com/attachments/7087https://www.physicsforums.com/attachments/7091The same conditions from Stone-Weierstrass apply.
Sturn-Liouville theory is a mathematical theory that deals with the study of ordinary differential equations, particularly those that involve boundary value problems. It was developed by mathematicians Jacques Charles François Sturm and Joseph Liouville in the 19th century.
The key concepts in Sturn-Liouville theory include the Sturm-Liouville operator, which is a second-order differential operator, and the associated eigenvalue problem. The theory also involves concepts such as eigenfunctions, eigenvalues, and orthogonality.
Sturn-Liouville theory has many applications in physics, engineering, and other fields. It is used to solve problems involving heat transfer, diffusion, and wave equations. It also has applications in quantum mechanics, signal processing, and fluid dynamics.
The main properties of the Sturm-Liouville operator include self-adjointness, which means that it is equal to its own adjoint, and the existence of a complete set of eigenfunctions. It also has a discrete spectrum of eigenvalues and satisfies a specific orthogonality condition.
Sturn-Liouville theory is closely related to other mathematical theories such as Fourier analysis, spectral theory, and functional analysis. It also has connections to other areas of mathematics such as group theory, differential geometry, and linear algebra.