Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. (English) Zbl 0609.47001
Let \({\mathcal H}\) be a Hilbert space, A be a selfadjoint operator with simple spectrum and let \(P=(\phi,.)\phi\) for some fixed \(\phi\in {\mathcal H}\). Let \(A_{\lambda}=A+\lambda P\). Necessary and sufficient conditions are found for the following statements to hold true (a.e. means a.e. w.r.t. the Lebesgue measure):
a) \(A_{\lambda}\) has empty singular continuous spectrum for a.e. \(\lambda\) ;
b) \(A_{\lambda}\) has only point spectrum for a.e. \(\lambda\) ;
c) \(A_{\lambda}\) has only point spectrum in a fixed interval (a,b) for a.e. \(\lambda\).
These results are connected with the theory of random Hamiltonians and some important properties of the latter are proved. The Anderson model is considered with more details.
a) \(A_{\lambda}\) has empty singular continuous spectrum for a.e. \(\lambda\) ;
b) \(A_{\lambda}\) has only point spectrum for a.e. \(\lambda\) ;
c) \(A_{\lambda}\) has only point spectrum in a fixed interval (a,b) for a.e. \(\lambda\).
These results are connected with the theory of random Hamiltonians and some important properties of the latter are proved. The Anderson model is considered with more details.
Reviewer: O.Enchev
MSC:
| 47A10 | Spectrum, resolvent |
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
Keywords:
Hilbert space; selfadjoint operator with simple spectrum; empty singular continuous spectrum; point spectrum; random Hamiltonians; Anderson modelReferences:
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