Singular Continuous Spectrum Is Generic

In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense Gδ . In the spectral analysis of various operators of mathematical physics, a key step, often the hardest, is to prove that the operator has no continuous singular spectrum, that is, that the spectral measures for the operators have only pure point and absolutely continuous parts. Examples are the absence of such spectrum for N -body Schrödinger operators [3,19] and for the one-dimensional random models [12,7,8,24,18]. Our goal here is to announce results that show that singular continuous spectrum is lying quite close to many operators by proving it is often generic in Baire sense. Detailed proofs and further results will appear in three papers: one for general operators [22], one for rank one perturbations [6], and one for almost periodic Schrödinger operators [23]. Precursors of our results include work on generic ergodic processes [15,21], on special energies for Schr̈odinger operators/Jacobi matrices [11,4,5]. Gordon [13,14] has independently (and presumably, before us) proven Theorem 5. His method of proof is very different from ours. Recall that the Baire category theorem implies that if X is a complete metric space, a countable intersection of dense Gδ is still a dense Gδ and if X is perfect, then any dense Gδ has uncountable intersection with any open ball. Our first two results are for one-body Schrödinger operators and for the “generic Anderson model.” 1 Permanent Address: IIMAS-UNAM, Apdo. Postal 20-726, Admon No. 20, 01000 Mexico D.F., Mexico. Research partially supported by DGAPA-UNAM and CONACYT. 2 Department of Mathematics, University of California, Irvine, CA 92717. 3 Division of Physics, Mathematics and Astronomy, 253-37, California Institute of Technology, Pasadena, CA 91125. 4 This material is based upon work supported by the National Science Foundation under Grant No. DMS-9207071. The Government has certain rights in this material. 5 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. To be submitted to Bull. Amer. Math. Soc.