About the Project

continuous spectra


(0.002 seconds)

3 matching pages

1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
§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 [ λ a , λ b ] ( 0 , ) , often called bands, when q ( x ) is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … …
2: 18.39 Applications in the Physical Sciences
The properties of V ( x ) determine whether the spectrum, this being the set of eigenvalues of , is discrete, continuous, or mixed, see §1.18. … Such a superposition yields continuous time evolution of the probability density | Ψ ( x , t ) | 2 . … 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. … 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 ± 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). …
3: Bibliography S
  • B. W. Shore and D. H. Menzel (1968) Principles of Atomic Spectra. John Wiley & Sons Ltd., New York.
  • B. Simon (1995) Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.
  • I. I. Sobelman (1992) Atomic Spectra and Radiative Transitions. 2nd edition, Springer-Verlag, Berlin.