# discrete spectrum

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## 3 matching pages

##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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► The analogous orthonormality is
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►The

*point, or discrete*spectrum of $T$ is then given by ${\bm{\sigma}}_{p}=\{{\lambda}_{0},{\lambda}_{1},\mathrm{\dots}\}$. … ► … ►###### 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, ${\bm{\sigma}}_{c}=[0,\mathrm{\infty})$, or, said more simply,*in the continuum*. …##### 2: 18.39 Applications in the Physical Sciences

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►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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Singularity analysis of generating functions.
SIAM J. Discrete Math. 3 (2), pp. 216–240.
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Monodromy- and spectrum-preserving deformations. I.
Comm. Math. Phys. 76 (1), pp. 65–116.
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From continuous to discrete Painlevé equations.
J. Math. Anal. Appl. 180 (2), pp. 342–360.
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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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