# continuous spectra

♦
3 matching pages ♦

(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 $[{\lambda}_{a},{\lambda}_{b}]\in (0,\mathrm{\infty})$, 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 $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18.
…
►Such a superposition yields continuous time evolution of the probability density ${|\mathrm{\Psi}(x,t)|}^{2}$.
…
►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 $\pm \mathrm{\infty}$ 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). …