Abstract
For an operator,A, with cyclic vector ϕ, we studyA+λP, whereP is the rank one projection onto multiples of ϕ. If [α,β] ⊂ spec (A) andA has no a.c. spectrum, we prove thatA+λP has purely singular continuous spectrum on (α,β) for a denseG δ of λ's.
Similar content being viewed by others
References
Aronszajn, N.: On a problem of Weyl in the theory of singular Sturm-Liouville equations. Am. J. Math.79, 597–610 (1957)
Carmona, R.: Exponential localization in one-dimensional disordered systems. Duke Math. J.49, 191–213 (1982)
Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Boston: Birkhäuser, 1990
del Rio, R.: A forbidden set for embedded eigenvalues. To appear Proc. AMS
Donoghue, W.: On the perturbation of spectra. Comm. Pure Appl. Math.18, 559–579 (1965)
Goldsheid, I.: Asymptotics of the product of random matrices depending on a parameter. Soviet Math. Dokl.16, 1375–1379 (1975)
Goldsheid, I., Molchanov, S., Pastur, L.: A pure point spectrum of the stochastic onedimensional Schrödinger equation. Func. Anal. Appl.11, 1–10 (1977)
Gordon, A.: On exceptional value of the boundary phase for the Schrödinger equation of a half-line. Russ. Math. Surv.47, 260–261 (1992)
Gordon, A.: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys., to appear
Kirsch, W., Molchanov, S., Pastur, L.: One dimensional Schrödinger operators with high potential barriers. Operator Theory: Advances and Applications, Vol. 57, Basel: Birkhäuser, 1992, pp. 163–170
Kotani, S.: Lyapunov exponents and spectra for one-dimensional random Schrödinger operators. Contemp. Math.50, 277–286 (1986)
Simon, B.: Operators with singular continuous spectrum, I. General operators. Ann Math., to appear
Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Comm. Pure Appl. Math.39, 75–90 (1986)
Stolz, G.: Bounded solutions and absolute continuity of Sturm-Liouville operators, J. Math. Anal. Appl.169, 210–228 (1992)
Author information
Authors and Affiliations
Additional information
Communicated by T. Spencer
Research partially supported by DGAPA-UNAM and CONACYT.
This material is based upon work suported by the National Science Foundation under Grant No. DMS-9207071. The Government has certain rights in this material.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material.
Rights and permissions
About this article
Cite this article
Del Rio, R., Makarov, N. & Simon, B. Operators with singular continuous spectrum: II. Rank one operators. Commun.Math. Phys. 165, 59–67 (1994). https://doi.org/10.1007/BF02099737
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02099737