The Spectrum of an Adelic Markov Operator

TL;DR

This paper studies the spectral properties of an adelic Markov operator derived from SL(2,Z) representations, revealing a spectral gap on the real line and a circular spectrum in the complex plane.

Contribution

It introduces a novel Markov operator based on adelic representations of SL(2,Z) and analyzes its spectral structure in detail.

Findings

Real spectrum has a gap.

Non-real spectrum forms a circle of radius 1/√2.

Spectral properties differ between real and complex components.

Abstract

With the help of the representation of SL(2,Z) on the rank two free module over the integer adeles, we define the transition operator of a Markov chain. The real component of its spectrum exhibits a gap, whereas the non-real component forms a circle of radius 1/\sqrt{2}.