# singular continuous spectra

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## 1—10 of 153 matching pages

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

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###### §1.18(v) Point Spectra and Eigenfunction Expansions

… ►###### §1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases

… ►###### §1.18(vii) Continuous Spectra: More General Cases

… ►Should $q(x)$ be bounded but*random*, leading to*Anderson localization*, the spectrum could range from being*a dense point spectrum*to being*singular continuous*, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). … …##### 2: 31.13 Asymptotic Approximations

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►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999).
►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

##### 3: 1.10 Functions of a Complex Variable

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►If $f(z)$ is analytic within a simple closed contour $C$, and continuous within and on $C$—except in both instances for a finite number of singularities within $C$—then
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►(b) By specifying the value of $F(z)$ at a point ${z}_{0}$ (not a branch point), and requiring $F(z)$ to be continuous on any path that begins at ${z}_{0}$ and does not pass through any branch points or other singularities of $F(z)$.
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##### 4: Bibliography S

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Principles of Atomic Spectra.
John Wiley & Sons Ltd., New York.
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Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective.
Numer. Math. 98 (2), pp. 371–387.
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A simple approach to asymptotic expansions for Fourier integrals of singular functions.
Appl. Math. Comput. 216 (11), pp. 3378–3385.
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Operators with Singular Continuous Spectrum: I. General Operators.
Annals of Mathematics 141 (1), pp. 131–145.
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Atomic Spectra and Radiative Transitions.
2nd edition, Springer-Verlag, Berlin.
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##### 5: 18.39 Applications in the Physical Sciences

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►The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18.
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►Brief mention of non-unit normalized solutions in the case of mixed spectra appear, but as these solutions are not OP’s details appear elsewhere, as referenced.
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►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*. … ►Namely for fixed $l$ the infinite set labeled by $p$ describe only the ${L}^{2}$*bound states*for that single $l$, omitting the*continuum*briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii). … ►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as $x\to -1-$. …##### 6: 1.4 Calculus of One Variable

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###### §1.4(ii) Continuity

… ► ►A*removable singularity*of $f(x)$ at $x=c$ occurs when $f(c+)=f(c-)$ but $f(c)$ is undefined. … … ►###### Absolutely Continuous Stieltjes Measure

…##### 7: 2.7 Differential Equations

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###### §2.7(i) Regular Singularities: Fuchs–Frobenius Theory

… ►Other points ${z}_{0}$ are*singularities*of the differential equation. …All other singularities are classified as*irregular*. … ►###### §2.7(ii) Irregular Singularities of Rank 1

… ►Thus a regular singularity has rank 0. …##### 8: Bibliography

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Unsteady lifting-line theory as a singular-perturbation problem.
J. Fluid Mech 153, pp. 59–81.
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Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions.
J. Mathematical Phys. 16, pp. 961–970.
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Singularities of Differentiable Maps. Vol. II.
Birkhäuser, Boston-Berlin.
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Normal forms of functions near degenerate critical points, the Weyl groups ${A}_{k},{D}_{k},{E}_{k}$ and Lagrangian singularities.
Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
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Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices.
Bulletin of the American Mathematical Society 6 (1), pp. 81–85.
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##### 9: 31.12 Confluent Forms of Heun’s Equation

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►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity.
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►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty}$.
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►This has irregular singularities at $z=0$ and $\mathrm{\infty}$, each of rank $1$.
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►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty}$ of rank $2$.
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►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty}$.
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##### 10: 31.14 General Fuchsian Equation

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►The general second-order

*Fuchsian equation*with $N+1$ regular singularities at $z={a}_{j}$, $j=1,2,\mathrm{\dots},N$, and at $\mathrm{\infty}$, is given by ►
31.14.1
$$\frac{{d}^{2}w}{{dz}^{2}}+\left(\sum _{j=1}^{N}\frac{{\gamma}_{j}}{z-{a}_{j}}\right)\frac{dw}{dz}+\left(\sum _{j=1}^{N}\frac{{q}_{j}}{z-{a}_{j}}\right)w=0,$$
${\sum}_{j=1}^{N}{q}_{j}=0$.

►The exponents at the finite singularities
${a}_{j}$ are $\{0,1-{\gamma}_{j}\}$ and those at $\mathrm{\infty}$ are $\{\alpha ,\beta \}$, where
…The three sets of parameters comprise the *singularity parameters*${a}_{j}$, the*exponent parameters*$\alpha ,\beta ,{\gamma}_{j}$, and the $N-2$ free*accessory parameters*${q}_{j}$. … ►
31.14.3
$$w(z)=\left(\prod _{j=1}^{N}{(z-{a}_{j})}^{-{\gamma}_{j}/2}\right)W(z),$$

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