[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/sturm-liouville-theory-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/sturm-liouville-theory-wikipedia\/","headline":"Sturm\u2013Liouville theory – Wikipedia","name":"Sturm\u2013Liouville theory – Wikipedia","description":"before-content-x4 Mathematical theory after-content-x4 In mathematics and its applications, classical Sturm\u2013Liouville theory is the theory of real second-order linear ordinary","datePublished":"2019-09-04","dateModified":"2019-09-04","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b7ed22ff2571ca9f5bc6f1236c5f57aea09ec786","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b7ed22ff2571ca9f5bc6f1236c5f57aea09ec786","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/sturm-liouville-theory-wikipedia\/","wordCount":21673,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Mathematical theory       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In mathematics and its applications, classical Sturm\u2013Liouville theory is the theory of real second-order linear ordinary differential equations of the form:ddx[p(x)dydx]+q(x)y=\u2212\u03bbw(x)y,{displaystyle {frac {d}{dx}}!!left[,p(x){frac {dy}{dx}}right]+q(x)y=-lambda ,w(x)y,}(1)for given coefficient functions  p(x), q(x), and w(x), an unknown function y = y(x) of the free variable x, and an unknown constant \u03bb. All homogeneous (i.e. with the right-hand side equal to zero) second-order linear ordinary differential equations can be reduced to this form. In addition, the solution y is typically required to satisfy some boundary conditions at extreme values of x. Each such equation (1) together with its boundary conditions constitutes a Sturm\u2013Liouville problem.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In the simplest case where all coefficients are continuous on the finite closed interval [a, b] and p has continuous derivative, a function y = y(x) is called a solution if it is continuously differentiable and satisfies the equation (1) at every x\u2208(a,b){displaystyle xin (a,b)}. In the case of more general p(x), q(x), w(x), the solutions must be understood in a weak sense.The function w(x), sometimes denoted r(x), is called the weight or density function.The value of \u03bb is not specified in the equation: finding the \u03bb for which there exists a non-trivial solution is part of the given Sturm\u2013Liouville problem. Such values of \u03bb, when they exist, are called the eigenvalues of the problem, and the corresponding solutions are the eigenfunctions associated to each \u03bb. This terminology is because the solutions correspond to the eigenvalues and  eigenfunctions of a Hermitian differential operator in an appropriate Hilbert space of functions with inner product defined using the weight function. Sturm\u2013Liouville theory studies the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in the function space.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4This theory is important in applied mathematics, where Sturm\u2013Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations. For example, in quantum mechanics, the one-dimensional time-independent Schr\u00f6dinger equation is a Sturm\u2013Liouville problem.A Sturm\u2013Liouville problem is said to be regular if p(x), w(x) > 0, and p(x), p\u2032(x), q(x), w(x) are continuous functions over the finite interval [a,b], and the problem has separated boundary conditions of the form:"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/sturm-liouville-theory-wikipedia\/#breadcrumbitem","name":"Sturm\u2013Liouville theory – Wikipedia"}}]}]