Sturm-Liouville form

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1: 1.13 Differential Equations

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes $\frac{d}{dx}$. … ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

2: 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. … ►

The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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3: 18.36 Miscellaneous Polynomials

… ►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the $L_n^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness. … ►Exceptional type I $X_m$-EOP’s, form a complete orthonormal set with respect to a positive measure, but the lowest order polynomial in the set is of order $m$, or, said another way, the first $m$ polynomial orders, $0,1,\mathrm{\dots },m-1$ are missing. … ►The $y(x)={\widehat{L}}_n^{(k)}\left(x\right)$ satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients: … ►In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …

4: 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 … ►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. …

5: Bibliography G

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An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 359 (1), pp. 352–367.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Li Zhong Peng)

Referenced by:

§18.36(vi).

Encodings:

BibTeX

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The evaluation of integrals of the form ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-x^2}𝑑x$ . Proc. Cambridge Philos. Soc. 45 (2), pp. 241–245.

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External Links:

MathReview (S. Levy), ZentralBlatt

Referenced by:

§3.5(i).

Encodings:

BibTeX

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General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.

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External Links:

Document

Referenced by:

item Greene et al. (1979):, Notations F, Notations F, Notations G.

Encodings:

BibTeX

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6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ► … ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. …Friedman (1990) provides a useful introduction to both approaches; as does the conference proceeding Amrein et al. (2005), overviewing the combination of Sturm–Liouville theory and Hilbert space theory. See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of $51$ solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.

7: 18.39 Applications in the Physical Sciences

… ►The orthonormal stationary states and corresponding eigenvalues are then of the form … ►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with $L^2$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty }$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …, $k=1$, there might be 3 nodes, rather than Sturm’s 1. …Both satisfy Sturm’s theorem. … ►These, taken together with the infinite sets of bound states for each $l$, form complete sets. …