On Last’s intersection spectrum conjecture

Abstract

In this paper, we prove Last's intersection spectrum conjecture in the measure sense for sufficiently smooth potentials. More precisely, we consider discrete one-dimensional quasi-periodic Schrödinger operators H v,α,θ with sufficiently smooth potentials and irrational α. Let S − ( pn qn ) denote the intersection of the spectra of H v, pn qn ,θ taken over θ where pn qn is the continued fraction expansion of α. We show that almost everywhere in α, up to sets of zero Lebesgue measure, the zero Lyapunov exponent regime is the limit of S − ( pn qn ).

Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

“…More recently, the conjecture was settled for a.e. α and sufficiently smooth potential by Zhao [Zha19].…”