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  "body": "\n## Definition\n\nLet $-\\infty \\le a \u003c b \\le \\infty$.\nLet $p, q, w \\colon (a, b) \\to \\R$ be measurable with $p \u003e 0$ and $w \u003e 0$ almost everywhere, and with $1/p$, $q$ and $w$ locally integrable on $(a, b)$.\nFor $\\lambda \\in \\C$ consider the [[Sturm–Liouville|7WV6B2502K]] equation\n$$-(p y')' + q y = \\lambda w y \\quad \\text{on } (a, b).$$\nIts solutions are the functions $y$ such that $y$ and $p y'$ are locally absolutely continuous and the equation holds almost everywhere.\nFix $c \\in (a, b)$.\nThe endpoint $b$ is in the limit-circle case if there is a $\\lambda \\in \\C$ such that every solution $y$ of the equation satisfies\n$$\\int_c^b \\abs{y(x)}^2 w(x) \\, dx \u003c \\infty.$$\nOtherwise $b$ is in the limit-point case.\nThe same definition applies to the endpoint $a$ with the interval $(a, c)$.\n\n## Remarks\n\nThe definition does not depend on $c$.\nWeyl's alternative says that if every solution is square integrable near $b$ for one $\\lambda \\in \\C$, then this holds for every $\\lambda \\in \\C$.\nFor $\\lambda \\in \\C \\setminus \\R$ there is always at least one nonzero solution that is square integrable near $b$.\nSo $b$ is limit-point exactly when, for such $\\lambda$, this solution is unique up to a constant factor.\nExample: for $p = w = 1$, $q = 0$ on $(0, \\infty)$ and $\\lambda = 0$, the solutions are $y = c_1 + c_2 x$, and $y = 1$ is not square integrable near $\\infty$.\nBy Weyl's alternative, $\\infty$ is in the limit-point case.\nNon-example of limit-point: a regular endpoint, such as $b \u003c \\infty$ with $1/p$, $q$, $w$ integrable near $b$, is always in the limit-circle case, since every solution then extends continuously to $b$.\nFor $p = w = 1$ and $q(x) = \\kappa / x^2$ on $(0, 1]$, the endpoint $0$ is in the limit-point case exactly when $\\kappa \\ge 3/4$.\nIf $p = w = 1$ and $q$ is bounded below near $b = \\infty$, then $\\infty$ is in the limit-point case.\n\n## Explanations\n\n### L1 human\n\nLook at all solutions near the endpoint and ask whether they are square integrable there.\nIf all of them are, the endpoint is limit-circle; if some are not, it is limit-point.\n",
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