Operators with singular continuous spectrum: II. Rank one operators

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.

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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.

Communicated by T. Spencer

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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

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Keywords

  • Neural Network
  • Statistical Physic
  • Complex System
  • Nonlinear Dynamics
  • Quantum Computing