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GENERIC SINGULAR SPECTRUM FOR ERGODIC SCHRODINGER OPERATORS
 

Summary: GENERIC SINGULAR SPECTRUM FOR ERGODIC
SCHR¨ODINGER OPERATORS
ARTUR AVILA AND DAVID DAMANIK
Abstract. We consider Schr¨odinger operators with ergodic potential V(n) =
f(T n()), n Z, , where T : is a non-periodic homeomorphism.
We show that for generic f C(), the spectrum has no absolutely continuous
component. The proof is based on approximation by discontinuous potentials
which can be treated via Kotani Theory.
1. Introduction
Let be a compact metric space, T : a homeomorphism, and µ a T-
ergodic Borel measure. We will always assume that T is not periodic, that is, µ is
non-atomic. For a bounded and measurable function f : R, we consider (line)
Schr¨odinger operators H = +V, n Z with potential V(n) = f(Tn
) and the
associated Lyapunov exponents (z), z C. By Kunz-Souillard (cf. [5, 12]), there
exists a compact set ac(f) R such that ac(H) = ac(f) for µ-a.e. . By
Pastur-Ishii-Kotani (cf. [5, 9, 10, 14, 15]), ac(f) = {E R : (E) = 0}
ess
.
We shall only consider situations where the potentials V are not periodic. In this

  

Source: Avila, Artur - Instituto Nacional de Matemática Pura e Aplicada (IMPA)

 

Collections: Mathematics