Purely Singular Continuous Spectrum for CMV Operators Generated by Subshifts

Abstract

We prove uniform absence of point spectrum for CMV operators corresponding to the period doubling subshift. We also prove almost sure absence of point spectrum for CMV operators corresponding to a class of Sturmian subshifts. Lastly, we prove almost sure absence of point spectrum for CMV operators corresponding to some subshifts generated by a coding of a rotation.

Purely singular continuous spectrum for Sturmian CMV matrices via strengthened Gordon Lemmas

“…Consequently, the new content of Theorem 1.1 is supplied by proving that E ω does not have eigenvalues. Let us note that this extends the results of [30] in two directions. First, we do not need to restrict the continued fraction coefficients of the frequency θ. Secondly, [30] makes a statement for Lebesgue-almost every phase ϕ ∈ [0, 1), while this result describes the spectral type for arbitrary phase.…”

“…Notice that Γ < ∞ (and hence ∆ > 0), since α is bounded away from ∂D. Moreover, we also have Γ ≥ 1 (and hence ∆ ≤ η) since det(S(α, z 0 )) = z 0 for all α ∈ D. If n k is even, one may apply the arguments from [30] without modification to deduce that max( Φ(2n k ) , Φ(n k ) ) = max( Φ(2n k ) , Φ(n k ) ) ≥ η ≥ ∆, since (1.8) implies that |tr(Z(j, 0; z))| = |tr(T (j, 0; z))| whenever j is even and z ∈ ∂D. Now, suppose n k is odd.…”

“…In general, one hopes to study the direct spectral problem for such matrices and prove theorems that establish the spectral type for a class of sequences of Verblunsky coefficients -for example, periodic coefficients always have purely absolutely continuous spectral type, and random coefficients (almost surely) have purely pure point spectral type. 1 Recently, aperiodic dynamically defined models for which α n assumes only finitely many values have been studied [1,7,8,14,15,16,30]. These models are popular as they furnish mathematical models of one-dimensional quasicrystals.…”

“…This philosophy was partially extended to CMV matrices by Ong in [28,29,30], with the caveat that one could only use scales of repetition of even length. At the time, this appeared to be an intrinsic drawback to CMV matrices, arising from the readily apparent fact that even and odd Verblunsky coefficients are not on precisely equal footing vis-à-vis the generalized eigenvalue equation Eψ = zψ, an asymmetry that is manifested in the alternating structure of the Gesztesy-Zinchenko transfer matrices (which propagate solution data for the eigenvalue equation and will be defined presently).…”

“…where u n is defined by (1.3). Our theorem for these is precisely [30,Theorem 4] with the hypothesis that θ have infintely many even continued fraction denominators removed.…”