On the Kotani-Last and Schrodinger conjectures

TL;DR

This paper disproves two longstanding conjectures in ergodic one-dimensional Schrödinger operators by demonstrating that ac spectrum rigidity can be broken through slow deformation of periodic potentials.

Contribution

It introduces a method of slow deformation of periodic potentials to disprove the conjectures linking ac spectrum and almost periodicity, challenging previous assumptions.

Findings

Disproved the conjecture that ac spectrum requires almost periodic potentials.

Showed that eigenfunctions can be unbounded in the ac spectrum.

Demonstrated that ac spectrum rigidity can be broken by slow deformation.

Abstract

In the theory of ergodic one-dimensional Schrodinger operators, ac spectrum has been traditionally expected to be very rigid. Two key conjectures in this direction state, on one hand, that ac spectrum demands almost periodicity of the potential, and, on the other hand, that the eigenfunctions are almost surely bounded in the essential suport of the ac spectrum. We show how the repeated slow deformation of periodic potentials can be used to break rigidity, and disprove both conjectures.