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  "body": "\n## Definition\n\nLet $q \\colon [0, \\infty) \\to \\R$ be integrable on each interval $[0, R]$ with $R \u003e 0$.\nAssume that the [[Schrödinger operator|89AA6XJY5K]] $-y'' + q y$ is in the [[limit-point case|6J9FXT6HAV]] at $\\infty$.\nFor $z \\in \\C \\setminus \\R$, let $\\psi(z, \\cdot)$ be a nonzero solution of $-y'' + q y = z y$ with $\\int_0^\\infty \\abs{\\psi(z, x)}^2 \\, dx \u003c \\infty$; by the limit-point assumption it is unique up to a constant factor.\nThe Weyl–Titchmarsh function, or $m$-function, of $q$ is\n$$m(z) = \\frac{\\psi'(z, 0)}{\\psi(z, 0)}, \\qquad z \\in \\C \\setminus \\R.$$\nWeyl–Titchmarsh theory is the study of the spectral theory of such operators through the function $m$.\n\n## Remarks\n\nThe quotient does not depend on the choice of $\\psi$.\nAlso $\\psi(z, 0) \\ne 0$, since otherwise the non-real number $z$ would be an eigenvalue of the self-adjoint operator with the Dirichlet condition $y(0) = 0$.\nThe function $m$ is analytic on $\\C \\setminus \\R$, satisfies $m(\\overline{z}) = \\overline{m(z)}$, and maps the [[upper half-plane|X6XGGH5FPW]] into itself.\nSo $m$ is a Herglotz function, and the measure in its [[Herglotz representation|9ZBAX4ZPX0]] is the spectral measure of the operator with the condition $y(0) = 0$.\nExample: for $q = 0$, the solution $\\psi(z, x) = e^{i\\sqrt{z}\\, x}$, with the branch of $\\sqrt{z}$ that has positive imaginary part, is square integrable on $[0, \\infty)$.\nSo $m(z) = i \\sqrt{z}$; for $z = i$ this is $e^{3\\pi i/4}$, which lies in the upper half-plane.\nNon-example: for $z \\in \\C \\setminus \\R$ the solution $e^{-i\\sqrt{z}\\, x}$ grows exponentially, so it cannot be used for $\\psi$.\nSome books use the other sign convention, or the boundary condition $y'(0) = h y(0)$ at $0$, which changes $m$ by a Möbius transformation.\n\n## Explanations\n\n### L1 human\n\nThe $m$-function compares the slope and the value at $0$ of the one solution that stays square integrable at infinity.\nIt is a single analytic function, and its boundary behaviour on the real line encodes the spectrum of the operator.\n",
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