discrete spectrum

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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► The analogous orthonormality is … ►The point, or discrete spectrum of $T$ is then given by ${𝝈}_p=\{{\lambda }_0,{\lambda }_1,\mathrm{\dots }\}$. … ► … ►

Example 1: In one and two dimensions any $q(x)$ with a ‘Dip, or Well’ has a partly discrete spectrum

… ►The bound states are in the negative energy discrete spectrum, and the scattering states are in the positive energy continuous spectrum, ${𝝈}_c=[0,\mathrm{\infty })$, or, said more simply, in the continuum. …

2: 18.39 Applications in the Physical Sciences

… ►While non-normalizable continuum, or scattering, states are mentioned, with appropriate references in what follows, focus is on the $L^2$ eigenfunctions corresponding to the point, or discrete, spectrum, and representing bound rather than scattering states, these former being expressed in terms of OP’s or EOP’s. … ►However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of $\mathscr{H}$ form a discrete, normed, and complete basis for a Hilbert space. … ►Here are three examples of solutions for (18.39.8) for explicit choices of $V(x)$ and with the ${\psi }_n(x)$ corresponding to the discrete spectrum. … ►The spectrum is entirely discrete as in §1.18(v). … ►The spectrum is entirely discrete as in §1.18(v). …

3: Bibliography F

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Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv., 38(1992), no. 3, pp. 621–635 (1992)

External Links:

ISSN 0373-2436, Document, MathReview (Andreĭ Bolibrukh), ZentralBlatt

Referenced by:

§31.13.

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Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

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External Links:

ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§2.10(iv).

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Monodromy- and spectrum-preserving deformations. I. Comm. Math. Phys. 76 (1), pp. 65–116.

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External Links:

ISSN 0010-3616, Link, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(ii), §32.4(iii), §32.8(ii).

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

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Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.

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External Links:

ISSN 0010-3616, Link, MathReview (Nikolai A. Kostov), ZentralBlatt

Referenced by:

§32.15.

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