Operators with Singular Continuous Spectrum: I. General Operators

§0. Introduction The Baire category theorem implies that the family, F , of dense sets Gδ in a fixed metric space, X, is a candidate for generic sets since it is closed under countable intersections; and if X is perfect (has no isolated point), then A ∈ F has uncountable intersections with any open ball in X. There is a long tradition of soft arguments to prove that certain surprising sets are generic. For example in C[0, 1], a generic function is nowhere differentiable. Closer to our concern here, Zamfirescu [20] has proven that a generic monotone function has purely singular continuous derivative, and Halmos [7]-Rohlin [14] have proven that a generic ergodic process is weak mixing but not mixing. We will say a set S ⊂ X is Baire typical if it is a dense Gδ and a set S ⊂ X is Baire null if its complement is Baire typical. Our goal is to look at generic sets of self-adjoint operators and show that their spectrum is quite often purely singular continuous. Here are three of our results that give the flavor of what we will prove in §3 and §4. Consider the sequence space, [−a, a]Z, of sequences vn with |vn| ≤ a. Given any such v, we can define a Jacobi matrix J(v) as the tridiagonal matrix with Jn,n±1 = 1 and Jn,n = vn. View J as a self-adjoint operator on (Z). It is known (e.g. [4,17,16]) that if one puts a product of normalized Lebesgue measures on [−a, a] (i.e., the vn are independent random variables each uniformly distributed in [−a, a]) then, J(v) is a.e. an operator with spectrum [a−2, a+2] and the spectrum there is pure point. So our first result is somewhat surprising. Theorem 1. View [−a, a]Z in the product topology. Then {v | J(v) has spectrum [−a− 2, a+ 2] and the spectrum is purely singular continuous} is Baire typical. We also have some results if Z is replaced by Z and the Jacobi matrix by the multidimensional discrete Schrödinger operator. One might think that the weakness of the topology and the one dimension are critical. They are not, as our second result shows. For V ∈ C(R ν ), let S(V ) be the Schrödinger operator −∆+ V on L(R ).