# Liouville normal form

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## 6 matching pages

##### 1: 30.2 Differential Equations

##### 2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

##### 3: 1.13 Differential Equations

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###### Transformation to Liouville normal Form

►Equation (1.13.26) with $x\in [a,b]$ may be transformed to the*Liouville normal form*…##### 4: Errata

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►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions.
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##### 5: 3.7 Ordinary Differential Equations

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►The remaining two equations are supplied by boundary conditions of the form
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###### §3.7(iv) Sturm–Liouville Eigenvalue Problems

… ►The*Sturm–Liouville eigenvalue problem*is the construction of a nontrivial solution of the system …The eigenvalues ${\lambda}_{k}$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy … ►If $q(x)$ is ${C}^{\mathrm{\infty}}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. …##### 6: 18.39 Applications in the Physical Sciences

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►All are written in the same form as the product of three factors: the square root of a weight function $w(x)$, the corresponding OP or EOP, and constant factors ensuring unit normalization.
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►By Table 18.3.1#12 the normalized stationary states and corresponding eigenvalues are
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►There is no need for a normalization constant here, as appropriate constants already appear in §18.36(vi).
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►Orthogonality and normalization of eigenfunctions of this form is respect to the measure ${r}^{2}dr\mathrm{sin}\theta d\theta d\varphi $.
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►Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below.
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