# continuous spectrum

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## 5 matching pages

##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Eigenfunctions corresponding to the continuous spectrum are non-$L^{2}$ functions. …, $T$ has no eigenfunctions in $L^{2}\left(X\right)$, then the spectrum $\boldsymbol{\sigma}$ of $T$ consists only of a continuous spectrum, referred to as $\boldsymbol{\sigma}_{c}$. … and completeness relation … The bound states are in the negative energy discrete spectrum, and the scattering states are in the positive energy continuous spectrum, $\boldsymbol{\sigma}_{c}=[0,\infty)$, or, said more simply, in the continuum. …
##### 2: Bibliography
• J. Avron and B. Simon (1982) Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.
• ##### 3: Bibliography S
• B. Simon (1995) Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.
• ##### 4: 18.39 Applications in the Physical Sciences
The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathcal{H}$, is discrete, continuous, or mixed, see §1.18. … The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by $(\alpha\hbar\gamma)^{2}/(2m)$ ($\gamma\geq 0$) with corresponding eigenfunctions ${\mathrm{e}}^{\alpha(x-x_{e})/2}W_{\lambda,\mathrm{i}\gamma}\left(2\lambda{% \mathrm{e}}^{-\alpha(x-x_{e})}\right)$ expressed in terms of Whittaker functions (13.14.3). … For $Z>0$ these are the repulsive CP OP’s with $x\in[-1,1]$ corresponding to the continuous spectrum of $\mathcal{H}(Z)$, $\epsilon\in(0,\infty)$, and for $Z<0$ we have the attractive CP OP’s, where the spectrum is complemented by the infinite set of bound state eigenvalues for fixed $l$. …
##### 5: Bibliography F
• R. H. Farrell (1985) Multivariate Calculation. Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York.
• M. V. Fedoryuk (1991) Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).
• H. Flaschka and A. C. Newell (1980) Monodromy- and spectrum-preserving deformations. I. Comm. Math. Phys. 76 (1), pp. 65–116.
• A. S. Fokas, B. Grammaticos, and A. Ramani (1993) From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.
• A. S. Fokas, A. R. Its, and X. Zhou (1992) 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.