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GLOBAL THEORY OF ONE-FREQUENCY SCHRODINGER OPERATORS II: ACRITICALITY AND FINITENESS OF PHASE
 

Summary: GLOBAL THEORY OF ONE-FREQUENCY SCHRšODINGER
OPERATORS II: ACRITICALITY AND FINITENESS OF PHASE
TRANSITIONS FOR TYPICAL POTENTIALS
ARTUR AVILA
Abstract. We consider Schršodinger operators with a one-frequency analytic
potential. Energies in the spectrum can be classified as subcritical, critical or
supercritical, by analogy with the almost Mathieu operator. Here we show that
the critical set is empty for an arbitrary frequency and almost every potential.
Such acritical potentials also form an open set, and have several interesting
properties: only finitely many "phase transitions" may happen, however never
at any specific point in the spectrum, and the Lyapunov exponent is minorated
in the region of the spectrum where it is positive.
1. Introduction
This work continues the global analysis of one-dimensional Schršodinger operators
with an analytic one-frequency potential started in [A1], to which we refer the reader
for further motivation.
For R Q and v C
(R/Z, R), let H = H,v be the Schršodinger operator
(1) (Hu)n = un+1 + un-1 + v(n)un
on 2

  

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

 

Collections: Mathematics