Liouville normal form

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1: 30.2 Differential Equations

… ►The Liouville normal form of equation (30.2.1) is …

2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ► … ► …

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

… ►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. …

5: 3.7 Ordinary Differential Equations

… ►The remaining two equations are supplied by boundary conditions of the form … ►

§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

… ►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. … ►By Table 18.3.1#12 the normalized stationary states and corresponding eigenvalues are … ►There is no need for a normalization constant here, as appropriate constants already appear in §18.36(vi). … ►Orthogonality and normalization of eigenfunctions of this form is respect to the measure $r^{2}dr\sin \theta d\theta d\varphi$. … ►Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below. …