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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers.\nLet $p \\in C^1([a, b])$ and $q, w \\in C([a, b])$ be real functions with $p(x) \u003e 0$ and $w(x) \u003e 0$ for all $x \\in [a, b]$.\nLet $\\alpha_1, \\alpha_2, \\beta_1, \\beta_2$ be real numbers with $(\\alpha_1, \\alpha_2) \\ne (0, 0)$ and $(\\beta_1, \\beta_2) \\ne (0, 0)$.\nThe regular Sturm–Liouville problem with these data is the [[boundary value problem|TM8VQJWGHZ]]\n$$-(p y')' + q y = \\lambda w y \\quad \\text{on } [a, b],$$\n$$\\alpha_1 y(a) + \\alpha_2 y'(a) = 0, \\qquad \\beta_1 y(b) + \\beta_2 y'(b) = 0,$$\nwhere $\\lambda$ is a complex parameter.\nA number $\\lambda$ is an eigenvalue of the problem if the problem has a solution $y \\in C^2([a, b])$ that is not identically zero.\nSuch a $y$ is an eigenfunction for $\\lambda$.\nSturm–Liouville theory is the study of the eigenvalues and eigenfunctions of such problems.\nIt also studies singular versions, where the interval is unbounded or the coefficients are singular at an end.\n\n## Remarks\n\nExample: $p = w = 1$, $q = 0$ on $[0, \\pi]$ with $y(0) = y(\\pi) = 0$.\nThen $y = \\sin(nx)$ satisfies $-y'' = n^2 y$ and both boundary conditions, so $n^2$ is an eigenvalue for each $n \\ge 1$.\nNon-example: the problem $-y'' = \\lambda y$ with $y(0) = y(2\\pi)$, $y'(0) = y'(2\\pi)$ is not a regular Sturm–Liouville problem in this sense.\nIts [[periodic boundary conditions|3FW4J270FV]] are not separated, and the eigenvalue $1$ then has the two eigenfunctions $\\cos x$ and $\\sin x$.\nWith the inner product $\\langle f, g \\rangle = \\int_a^b f \\overline{g}\\, w \\, dx$, the operator $y \\mapsto \\frac{1}{w}(-(p y')' + q y)$ with these boundary conditions is symmetric.\nThis is why the eigenvalues are real and eigenfunctions for distinct eigenvalues are orthogonal.\n\n## Explanations\n\n### L1 human\n\nA Sturm–Liouville problem is a second-order linear equation with a free number $\\lambda$ and one condition at each end.\nFor most values of $\\lambda$ only the zero function fits; the special values where a nonzero solution fits are the eigenvalues.\n",
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