Liouville normal form

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2020 Mathematics Subject Classification: Primary: 34A30 Secondary: 34B24 [MSN][ZBL]

$ \newcommand{deriv}[2]{\frac{\mathrm{d}#1}{\mathrm{d}#2}} \newcommand{derivn}[3]{\frac{\mathrm{d}^{#3}#1}{\mathrm{d}#2^{#3}}} $

The Liouville normal form is a way of writing a second-order ordinary linear differential equation \begin{equation}\label{eq1} \derivn{y}{x}{2} + p(x)\deriv{y}{x} + \left( q(x) + \lambda r(x) \right) y = 0, \end{equation} in the form \begin{equation}\label{eq2} \derivn{\eta}{\xi}{2} + \left( \lambda + \phi(\xi) \right) \eta = 0, \end{equation} where $\lambda$ is parameter. If $p(x) \in C^1$, $r(x) \in C^2$ and $r(x) > 0$, then equation \ref{eq1} reduces to the Liouville normal form \ref{eq2} by means of the substitution \[ \eta(\xi) = \Phi(x)y(x),\quad \xi = \int_\alpha^x \sqrt{r(t)}\,\mathrm{d}t, \quad \Phi(x) = r(x)^{1/4} \exp\left( \frac{1}{2}\int_\alpha^x p(t)\,\mathrm{d}t \right), \] which is called the Liouville transformation (introduced in [Li]). The Liouville normal form plays an important role in the investigation of the asymptotic behaviour of solutions of \ref{eq1} for large values of the parameter $\lambda$ or the argument, and in the investigation of the asymptotics of eigenfunctions and eigenvalues of the Sturm–Liouville problem (see [Ti]).


[In] E.L. Ince, "Ordinary differential equations", Dover, reprint (1956)
[Ka] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", 1. Gewöhnliche Differentialgleichungen, Chelsea, reprint (1947)
[Li] J. Liouville, J. Math. Pures Appl., 2 (1837) pp. 16–35
[Ti] E.C. Titchmarsh, "Eigenfunction expansions associated with second-order differential equations", 1–2, Clarendon Press (1946–1948)
How to Cite This Entry:
Liouville normal form. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article