continuous spectrum
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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Eigenfunctions corresponding to the continuous spectrum are non- functions.
…, has no eigenfunctions in , then the spectrum
of consists only of a continuous spectrum, referred to as .
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►and completeness relation
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►The bound states are in the negative energy discrete spectrum, and the scattering states are in the positive energy continuous spectrum, , or, said more simply, in the continuum.
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2: Bibliography
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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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3: Bibliography S
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Operators with Singular Continuous Spectrum: I. General Operators.
Annals of Mathematics 141 (1), pp. 131–145.
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4: 18.39 Applications in the Physical Sciences
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►The properties of determine whether the spectrum, this being the set of eigenvalues of , is discrete, continuous, or mixed, see §1.18.
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►The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by () with corresponding eigenfunctions expressed in terms of Whittaker functions (13.14.3).
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►For these are the repulsive CP OP’s with corresponding to the continuous spectrum of , , and for we have the attractive CP OP’s, where the spectrum is complemented by the infinite set of bound state eigenvalues for fixed .
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5: Bibliography F
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Multivariate Calculation. Use of the Continuous Groups.
Springer Series in Statistics, Springer-Verlag, New York.
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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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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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Continuous and Discrete Painlevé Equations.
In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.),
NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.
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