continuous spectra
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3 matching pages
1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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§1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases
… ►§1.18(vii) Continuous Spectra: More General Cases
►More generally, continuous spectra may occur in sets of disjoint finite intervals , often called bands, when is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … …2: 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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►Such a superposition yields continuous time evolution of the probability density .
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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 eigenfunctions vanishing at the end points, in this case see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem.
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►Namely for fixed the infinite set labeled by describe only the
bound states for that single , 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).
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